# Principles of Convection

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1 Principles of Convection

2 Point Conduction & convection are similar both require the presence of a material medium. But convection requires the presence of fluid motion. Heat transfer through the: Solid is always by conduction Liquid & gas can be by conduction or convection (depending on the presence of any bulk fluid motion)

3 Convection heat transfer is complicated involved fluid motion as well as heat conduction. The fluid motion enhances heat transfer (fluid velocity, the rate of heat transfer ).. Point Example of forced convection in daily life We resort to forced convection whenever we want to increased the rate of heat transfer from a hot object: 1) Turn on the fan on hot summer days to help our body cool more effectively (the higher the fan speed the better we feel) 2) Stir our soup & blow a hot slice of pizza to make them cool faster

4 Physical Mechanism Example Consider steady heat transfer through a fluid contained between two parallel plates at different temp. Assume no fluid motion, the energy of the hotter fluid molecules near the hot plate is transferred to the adjacent cooler fluid molecules. This energy is then transfer to the next layer of the cooler fluid molecules. This energy is then transferred to the next layer of the cooler fluid, and so on until it is transferred to the other plate. This is what happen during conduction through a fluid If we used a syringe to draw some fluid near the hot plate & inject it next to the cold plate repeatedly. So that, it will speed up the heat transfer process (energy is carried to the other side as a result of fluid motion) convection

5 Consider, the cooling of a hot block with a fan blowing air over its top surface We know that heat is transferred from the hot block to the surrounding cooler air, & the block eventually cools. The block cools faster if: 1) The fan is switched to higher speed 2) Replacing air by water The convection heat transfer strongly depends on: 1) Fluid properties dynamic viscosity (μ), thermal conductivity (k), density (ρ), specific heat (c p ) & fluid velocity (v) 2) The geometry & the roughness of the solid surface 3) The type of fluid flow (being streamlined or turbulent) Thus, the convection heat transfer relations to be rather complex because of the dependence on so many variables. The convection is the most complex mechanism of heat transfer.

6 Physical Mechanism Although the complexity of convection, the rate of convection is proportional to the temp. difference and expressed by Newton s Law of cooling as: or Where: The convection heat transfer coefficient, h defined as the rate of heat transfer between a solid surface & a fluid per unit surface area per unit temp. difference. The h depend on several of the mentioned variable and thus is difficult to determine.

7 Physical Mechanism Fluid flow is often restricted by solid surfaces. Important to understand how the presence of solid surface affects fluid flow. Consider the flow of fluid over a solid surface that is nonporous Observations indicate that a fluid in motion comes to complete stop at the surface and assume a zero velocity relative to the surface. That is, a fluid in direct contact with a solid stick to the surface due to viscous effects & there is no slip. This is known as the no-slip condition.

8 The layer that sticks to the surface slows the adjacent fluid layer because of viscous forces between the fluid layers, which slow the next layer. The flow region adjacent to the wall in which the viscous effects & thus velocity gradients are significant is called the boundary layer A consequence of the no-slip condition is: 1) All velocity profile must have zero value with respect to the surface at the point of contact between a fluid & solid surface 2) The surface drag which is the force a fluid exerts on a surface in the flow direction

9 Physical Mechanism An implication of the no-slip condition is that heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction since the fluid layer is motionless & can be expressed as: Heat is then convected away from a surface as a result of fluid motion. Note that convection heat transfer from a solid surface to a fluid is merely the conduction heat transfer from the solid surface to the fluid layer adjacent to the surface. So that, connect the equation for the heat flux to obtain: Used to get h when the temp. distribution within the fluid is known

10 Nusselt Number In convection studies, it is common practice to nondimensionalize the governing equations & combine the variables, which group together into dimensionless no. to reduce the no. of total variables. It is also common practice to nondimensionalize the heat transfer coefficient, h with Nusselt no., defined as: K: thermal conductivity of the fluid L c : Characteristic length

11 Nusselt Number The physical significance of the Nusselt no. Consider a fluid layer of thickness L & temp. difference Heat transfer through the fluid layer is by : 1) Convection (when fluid involved some motion) 2) Conduction (when the fluid layer is motionless) The ratio gives

12 Nusselt Number Represent the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer. Nu large the more effective the convection Nu = 1 heat transfer across the layer by pure conduction

13 Classification of Fluid Flows 1) Viscous versus Inviscid Regions of Flow # When 2 fluid layers move relatively to each other, a friction force develops between them & the slower layer tries to slow down the faster layer. This internal resistance to flow is quantified by the fluid property viscosity. # Viscosity is caused by cohesive forces between the molecules in liquids & by molecular collisions in gases. No fluid with zero viscosity. #Flow in which the frictional effects are significant are called viscous flows #Typically regions not close to solid surface, where viscous forces are negligible small compared to pressure forces. inviscid flow regions The development of viscous & inviscid regions of flow as a result of inserting a flat plate parallel into a fluid stream of uniform velocity

14 Classification of Fluid Flows 2) Internal versus External Flow Depending on whether the fluid if forced to flow: 1) In a pipe or duct (Internal Flow) fluid completely bounded by solid surface. Dominated by the influence of viscosity throughout the flow field. 2) Over a surface such as plate or pipe (External Flow) unbounded fluid. The viscous effects are limited to boundary layers near solid surfaces. 3) Compressible versus Incompressible Flow Depending on the level variation of density during flow. Incompressible if the flow density remains nearly constant & the volume of energy portion of fluids remain unchanged over the course of its motion. Gas flows as incompressible depends on the Mach number where c is the speed of sound, c=346 m/s

15 Classification of Fluid Flows 4) Laminar versus Turbulent Flow # Fluid motion characterized by smooth layer of the fluid Laminar layer # Disordered fluid motion typically occur at high velocities & is characterized by velocity fluctuations Turbulent Layer # The flow that alternates between being laminar & turbulent is called Transitional 5) Natural versus Force Flow # Depending on how the fluid motion is initiated 1) Forced flow fluid is forced to flow over a surface or in pipe by external means such as pump or fan 2) Natural flow fluid motion is due to natural means such as the buoyancy effect

16 Classification of Fluid Flows 6) Steady versus Unsteady Flow # Steady no change at a point with time # Unsteady change at a point with time # Uniform no change with location over a specified region.

17 1D, 2D and 3D Flows A flow field is best characterized by the velocity distribution & thus a flow is said to be 1D, 2D or 3D if the flow velocity varies in 1, 2 or 3 primary dimensions. Example a typical fluid flow involves a 3D geometry & the velocity may vary in all 3 dimensions, V(x,y,z) in rectangular. However, the variation of velocity in certain direction can be small relative to the variation on other directions & can be ignored.

18 1D, 2D and 3D Flows Consider steady flow of a fluid through a circular pipe attached to a large tank. The fluid velocity everywhere on the pipe surface is zero (because no-slip condition). The development of the velocity profile in a circular pipe, V=V(r,z) # The flow is 2D in the entrance region of the pipe since the velocity change in both r and z directions. The velocity profile develops fully & remains unchanged after some distance from the inlet and the flow in this region is said to be fully developed (1D since velocity just varies in the radial r- direction).

19 Velocity Boundary Layer Consider the parallel flow of a fluid over a flat plate. The x-coordinate measured along the plate surface from the leading edge of the plate in the direction of the flow. The y-coordinates measured from the surface in the normal direction. The development of the boundary layer for flow over a flat plate & the different flow regimes. # the fluid approaches the plate in the x-direction with a uniform velocity, V which is practically identical to the free stream velocity over the plate away from the surface.

20 Concept: Velocity Boundary Layer # the velocity of the particles in the first fluid layer adjacent to the plate become zero (because of the no-slip condition). This motionless layer slows down the particle of the neighboring fluid layers as a result of friction between the particles of these two adjoining fluid layers at different velocity. This fluid layer then slows down the molecules of the next layer and so on. Thus, the presence of the plate is felt up to some normal distance from the plate beyond which the free stream velocity remain essentially unchanged. As a result, the x-component of the fluid velocity, u varies from 0 at y=0 to V at y=. The region of the flow above the plate bounded by called velocity boundary layer. The boundary layer thickness, define as the distance y from the surface at which u=0.99v. The development of a boundary layer on a surface is due to the no-slip condition & friction

21 Wall shear Stress From the velocity boundary layer concept (the fluid layer in contact with the surface tries to drag the plate along via friction), apply a friction force on it. # Friction force per unit area is called Shear Stress, The shear stress at the wall surface is expressed as: Where: C f dimensionless friction coefficient. The value in most cases is determined experimentally. ρ density of the fluid. Note: Generally, the friction coefficient, C f varies with location along the surface. Once the average C f over a given surface is available, the friction force over the entire surface is determine The friction coefficient, C f important parameter in Heat Transfer studies since it is directly related to the Heat Transfer Coefficient

22 Thermal Boundary Layer A thermal boundary layer develops when a fluid at a specified temp. flows over a surface that is at different temp. Consider the flow of a fluid at a uniform temp. of T over an isothermal flat plate at temp. T s. These fluid particle then exchange energy with the particles in the adjoining fluid layer. The flow region over the surface in which the temp. variation in the direction normal to the surface is thermal boundary layer. Thermal boundary layer on a flat plate (the fluid is hotter than the plate surface) The thickness of thermal boundary layer, at any location along the surface is defined as the distance from the surface at which the temp. difference. Note: the fluid velocity has a strong influence on the temp. profile, the development of the velocity boundary layer relative to the thermal boundary layer will have a strong effect on the convection heat transfer.

23 Prandtl Number The relative thickness of the velocity & the thermal boundary layer is best described by the Prandtl no. (dimensionless parameter). The Prandtl numbers of fluids range from less than 0.01 for liquid metals to more than 100,000 for heavy oil.

24 Laminar & Turbulent Flow The flow regimes: 1) Laminar characterized by smooth stream-lines & highly-ordered motion 2) Transition Flow from laminar to turbulent occur over some region in which the flow fluctuates between laminar & turbulent 3) Turbulent characterized by velocity fluctuations & highly-disordered motion Most flows encountered in practice are turbulent. Laminar flows is encountered when highly viscous fluids such as oils flow in small pipes.

25 The turbulent boundary layer can be considered consists of 4 regions (characterized by the distance from the wall. 1) Viscous sublayer the very thin layer next to the wall where viscous effects are dominant. The velocity profile nearly linear & the flow is streamlined 2) Buffer layer the turbulent effects are becoming significant but the flow still dominated by viscous effects 3) Overlap layer the turbulent effects are much more significant but still not domain 4) Turbulent layer The turbulent effects dominate over viscous effects. # The intense mixing of the fluid in turbulent flow as result of rapid fluctuations enhances heat and momentum transfer between fluid particles, which increase the friction force on the surface & the convection heat transfer rate. # Both the friction & heat transfer coefficients reach maximum values when the flow become fully turbulent

26 Reynolds Number The flow regime depends on the ratio of the inertia forces to viscous forces in the fluid. This ratio is called Reynolds no. Where. V = upstream velocity (equivalent to the free-stream velocity for a flat plate) L c = characteristic length of the geometries = kinematic viscosity of the fluid (units: m 2 /s) # The critical Reynolds no. The Reynolds no. at which the flow become turbulent # The value of the critical Reynolds no. is different for different geometries and flow conditions. For flow over a flat plate, the general value of the critical Reynolds no is Where is the distance from the leading edge of the plate at which transition from laminar to turbulent flow occurs

27 Reynolds Number

28 Heat & Momentum Transfer in Turbulent Flow Concept: Most flows encountered in eng. practice are turbulence thus it is important to understand how turbulence affects wall shear stress & heat transfer. However, turbulence flow is a complex mechanism dominated by fluctuation. Turbulent flow characterized by disorderly & rapid fluctuations of swirling region of fluid called eddies. These fluctuation provide an additional mechanism for momentum & energy transfer. In laminar flow, fluid particles flow in an orderly manner along pathlines and momentum & energy are transferred across streamlines by molecular diffusion. In turbulent flow, the swirling eddies transport mass, momentum & energy to other regions of flow much more rapidly than molecular diffusion enhancing mass, momentum & heat transfer. As a result, turbulent flow is associated with much higher values of friction, heat transfer & mass transfer coefficient

29 Derivation of Differential Convection Equations Consider the parallel flow of a fluid over a surface. For analysis, take the flow direction along the surface to be x & the direction normal to the surface to be y and a differential volume element of length dx, height dy and unit depth in z-direction. The continuity, momentum and energy equations for steady 2D incompressible flow with constant properties are determined from mass. Momentum and energy balance to be: Where the viscous dissipation function,

30 Using the boundary layer approximations and a similarity variable, all the equation mention can be solved for parallel steady incompressible flow over a flat plate, with the following result The average friction coefficient and Nusselt no. are expressed in functional form as: and The Nusselt no. can be expressed by a simple power law relation of the form: m & n are constant exponents and the value of the constant, C depends on geometry.

31 The Reynolds analogy relates the convection coefficient to the friction coefficient for fluids with and is expressed as: Where; Stanton Number

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