Chapter 6. Energy Conservation. Islamic Azad University. Problems involving conduction: Chapters 2-3. Transient problems: Chapter 5

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1 Chapter 6 Introduction to Convection Islamic Azad University Karaj Branch 1 Energy Conservation Problems involving conduction: Chapters 2-3 Transient problems: Chapter 5 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 Obtained temperature as a function of time for cases where resistance to conduction was negligible 2

2 In Chapters 1-5 we used Newton s law of convection:! h was provided! we did not consider any temperature variations within the fluid 3 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. 4

3 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. 5 Heat Transfer Coefficient Recall Newton s law of cooling for heat transfer between a surface of arbitrary shape, area A s and temperature T s 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 (6.1) where (6.2) average heat transfer coefficient 6

4 Heat Transfer Coefficient For flow over a flat plate: (6.3)! How can we estimate the heat transfer coefficient? 7 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 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: and where C f is the local friction coefficient 8

5 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 The local heat flux is: (6.4) and (6.5) 9 Laminar and Turbulent Flow Transition criterion at Re critical : Transition criterion at Re critical : 10

6 Example Consider airflow over a flat plate of length L=1m under conditions for which transition occurs at x c =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? C lam =8.845 W/m 3/2.K C turb =49.75 W/m 1.8.K 11 Boundary Layers - Summary Velocity boundary layer (thickness (x)) characterized by the presence of velocity gradients and shear stresses - Surface friction, C f Thermal boundary layer (thickness t (x)) characterized by temperature gradients Convection heat transfer coefficient, h Concentration boundary layer (thickness c (x)) is characterized by concentration gradients and species transfer Convection mass transfer coefficient, h m 12

7 ! Need to determine the heat transfer coefficient, h (6.5)! Must know T(x,y), which depends on velocity field 13 Functional form of the solutions From dimensional analysis, or solution of boundary layer equations: (6.6) where Nu is the local Nusselt number (6.7) 14

8 Functional form of the solutions The average Nusselt number, based on the average heat transfer coefficient is: (6.8) where: Prandtl number Reynolds number 15 Physical meaning of dimensionless groups See Table 6.2 textbook for a comprehensive list of dimensionless groups 16

9 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. 17 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? 18

10 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 19