Convection Heat Transfer Department of Chemical Eng., Isfahan University of Technology, Isfahan, Iran Seyed Gholamreza Etemad Winter 2013 Heat convection: Introduction Difference between the temperature of the media and the fluid Energy transfer from a media to a fluid over it. Examples: Convective heat transfer occurs extensively in practice. 2
3 Convective heat transfer Heat Transfer Fluid Dynamics Introduction Forced Convection Free Convection Mixed Convection 4
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Introduction Question: From a conceptual viewpoint, is the convection heat transfer a basic mode of heat transfer? Several factors play major roles in convection heat transfer: (i)fluid motion (ii)fluid nature and properties (iii)surface geometry (iv)boundary conditions 7 The Continuum and Thermodynamic Equilibrium Concepts Focal Point in convection heat transfer temperature distribution in a moving fluid the determination of the T =T(x, y, z, t) Convection heat transfer depends on material properties such as density, pressure, thermal conductivity, and specific heat. In the continuum model the characteristics of individual molecules are ignored and average or macroscopic properties become important. A continuum is a media composed of continuous matter. It is valid for sufficiently large number of molecules in a given volume. Knudson number, Kn - Molecular Mean Free Path - the characteristic length, such as the equivalent diameter or the spacing between parallel plates. 8
The Continuum and Thermodynamic Equilibrium Concepts Continuum is valid for: Is continuum valid for micro and/or nano-channels? Thermodynamic equilibrium: fluid and the adjacent surface have the same velocity and temperature, no-velocity slip and notemperature jump. The condition for thermodynamic equilibrium: 9 Eulerian and Lagrangian Approaches There are two different points of view in analyzing problems in mechanics. 1- Eulerian method of description 2- Lagrangian method of description The Eulerian view, appropriate to fluid mechanics, is to specify the fluid properties (e.g. density, velocity) at each point in space at each instant of time. The density, for example, is then specified by a function: ρ = ρ x, y,z,t ( ) In the Euler picture, attention is focused on what is happening at a particular point in space, rather than on a particular fluid element. 10
Eulerian and Lagrangian Approaches Examples: --- When a pressure probe is introduced into a laboratory flow --- Analysis of traffic flow along a freeway --- Standing on a bridge and recording the variation of fish concentration below the bridge. The Lagrangian approach, more appropriate for solid mechanics, follows an individual particle moving through the flow. Suppose we have a fluid element that is at position (x o, y o, z o ) at time t o. At later times, the position of this element is described by functions: x=x(x o, y o, z o, t), y=y(x o, y o, z o, t), z=z(x o, y o, z o, t) Therefore, any field variable is given as: V=f[x(t), y(t), z(t), t] 11 Eulerian and Lagrangian Approaches These two descriptions are equivalent and there are relationships between the Lagrangian and Eulerian equations of fluid motion. Eulerian description, the time derivative is the partial derivative with respect to t keeping x, y, and z fixed. ρ = t ( x, y,z,t t ) ( x, y,z,t) lim ρ ρ t 0 In the Lagrangian method, the time derivative is the total derivative: dρ x x, y y,z z,t t x, y,z,t lim ρ = ρ = dt t 0 t dρ ρ ρ dx ρ dy ρ dz ρ ρ ρ ρ = = Vx Vy Vz dt t x dt y dt z dt t x y z Where V=(V x, V y, V z ) is the velocity of the fluid element. t ( ) ( ) 12
Eulerian and Lagrangian Approaches The relationship between the derivatives for any field variables (A) is: da A A A A A = Vx Vy Vz = ( V. A) dt t x y z t The operator d/dt is sometimes given a special name such as substantial derivative or material derivative and often assigned a special symbol such as D/Dt. Systems and Control Volumes: System is defined as an arbitrary quantity of mass of fixed identity. The Lagrangian method of fluid mechanics is used in the mathematical description of a system. 13 Eulerian and Lagrangian Approaches At a system, neglecting nuclear reactions, the quantity of mass is fixed. Thus the mass of the system is conserved and does not change. dm msyst = const, = 0 dt ---If the surroundings exert a net force F on the system, Newton s second law states that the mass will began to accelerate. dv d F = ma = m = mv dt dt ( ) In fluid mechanics Newton s law is called the conservation of linear momentum or alternately, the momentum principle. ---If heat Q is added to the system or work dw is done by the system, the system energy de must change according to the energy relation, or the first law of thermodynamics. dq dw de dq dw = de, = dt dt dt 14
Eulerian and Lagrangian Approaches Control Volume is the same as a system, except that the rest of the continuum may cross the fixed or deformable boundaries of the control volume at one or more places. This is the only difference between a control volume and a system. The Eulerian method of fluid mechanics is used in the mathematical description of a control volume. 15 Mass Conservation: For a control volume M t CV = mɺ mɺ inlet ports outlet ports 16
ρ x y z = y z ( ρvx ) ( ρv ) x x z ( ρvy ) ( ρvy ) x x x t y y y x y ( ρvz ) ( ρvz ) z z z By dividing both sides of the equation by and taking the limit as these dimensions approach zero, we get: ( x y z) ρ ρv ρv = x y ρvz t x y z This equation is called the continuity equation. We may write the continuity equation in vector form: ρ = = t Dt For a fluid of constant density: ( v). = 0 Dρ (. ρv ), ρ (. v) 17 Momentum Conservation: For a volume element x y z we write a momentum balance in this form: rate of rate of rate of sum of forces momentum = momentum momentum acting on accumulation in out system ( Mv ) n t CV = F mv ɺ mv ɺ n n n inlet outlet ports ports Where n is the direction chosen for analysis and (v n, F n ) are the projections of fluid velocity and forces on the n direction. This equation is the control volume formulation of Newton s second law of motion and is recognized in the literature as the momentum principle. 18
19 The convective flow of x-momentum must be considered across all six faces and that the net convective x-momentum flow into the volume element is: ( ρ x x ρ x x ) ( ρ x x x y x ρ y y x y y ) y z v v v v x z v v v v ( ρ z x ρ z x ) x y v v v v z z z The x-momentum by molecular transport: ( τ xx τ xx ) ( τ yx τ yx ) ( τ zx τ zx ) y z x z x y x x x y y y z z z 20
The forces related to pressure and gravity in x-direction will be: ( ) ρ x y z p p g x y z x x x The rate of accumulation of x-momentum within the element is: ρv x y z x t By dividing the entire resulting equation by x y z and taking the limit as x, y, and z of motion: approach zero, we obtain the x-component of the equation ρvx ρvxv ρv x yvx ρvzvx τ τ xx yx τ zx p = ρ g t x y z x y z x The y- and z-components are as following: ρvy ρvxvy ρvyvy ρvzvy τ xy τ yy τ zy p = ρ g t x y z x y z y 21 x y ρvz ρvxv ρv z yvz ρvzvz τ τ xz yz τ zz p = ρ g t x y z x y z z It is convenient to combine them to give the single vector equation: ρv = (. ρvv) (. τ ) p ρ g t ρdv = (. τ ) p ρ g Dt Where: v v x y vz τ xx = 2µ, τ yy = 2µ, τ zz = 2µ x y z v v x y vz vx τ xy = τ yx = µ, τ xz τ zx µ y x = = x z vy vz τ yz = τ zy = µ z y z 22
Euler equation: ρdv = p ρ g Dt Navier-Stokes equation: ρdv 2 = p ρ g µ v Dt 23 24
25 Energy Conservation: For a volume element x y z we write a momentum balance in this form: rate of rate of rate of accumulation internal and internal and = of internal and kinetic energy kinetic energy kinetic energy in by convection out by convection net rate of net rate of work done by heat addition system on by conduction surroundings 26
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The rate of accumulation of internal and kinetic energy within x y z is: 1 2 ρ e ρ u 2 x y z t Where e is the internal energy per unit mass of the fluid and u is the magnitude of the local velocity. The net rate of convection of internal and kinetic energy into the element is: 1 2 1 2 y z u ρ x e ρ u u ρ x e ρ u 2 x 2 x x 1 2 1 2 x z u ρ y e ρ u u ρ y e ρ u 2 y 2 y y 1 2 1 2 y x uz ρ e ρ u uz ρ e ρ u 2 z 2 z z 29 The net rate of energy input by conduction is: { x x } { y y } { z z } y z q q x z q q y x q q x x x y y y z z z Where q x, q y, q z are the x, y, and z components of the heat flux vector q. The work done by the fluid element against its surroundings consists of two parts: --- the work against the volume forces (body forces) e.g. gravity --- the work against the surface forces i.e. pressure and viscous forces Work = (Force) (Distance in the direction of the force) Rate of doing work = (Force) (Velocity in the direction of the force) The rate of doing work against the gravitational force per unit mass is: ( x x y y z z ) ρ x y z u g u g u g 30
The rate of doing work against the pressure at different faces is: { } {( x ) ( ) } ( ) ( ) x y y {( z ) ( z ) } y z pu pu x z pu pu y x pu pu x x x y y y z z z The rate of doing work against the viscous forces is: {( τ xx x τ xy y τ xz z ) ( τ xx x τ xy y τ xz z ) } x x x ( τ yx x τ yy y τ yz z ) ( τ yx x τ yy y τ yz z ) y z u u u u u u { y y } y {( τ zx x τ zy y τ zz z ) ( τ zx x τ zy y τ zz z ) } x z u u u u u u x y u u u u u u z z z By substituting the foregoing expressions into main energy equation and Dividing the entire equation by x y z while the dimensions approach zero The energy equation is obtained: 31 1 2 ρ e ρ u = t 2 1 2 1 2 1 2 u ρ e u u e u u e u x ρ ρ y ρ ρ z ρ x 2 y 2 z 2 q q x y q pu z pu x y puz ρ ( ux gx uy g y uz gz ) x y z x y z ( τ xxux τ xyuy τ xzuz ) ( τ yxux τ yyuy τ yzuz ) ( τ zxu ) x τ zyuy τ zzuz x y z In vector-tensor notation: 1 2 1 2 ρ e ρ u =. ρ u ρ e ρ u (.q ) ρ ( u.g) t 2 2 (.pu) (.( τ.u )) 32
33 Continuity Equation: 34
35 After simplification: For Newtonian fluids with constant density and thermal conductivity: 36
For Newtonian fluids with constant density and thermal conductivity: 37 For Newtonian fluids with constant density and thermal conductivity: 38
Reynolds Transport Theorem (R.T.T.) In order to convert a system analysis into a control volume analysis we must convert our mathematics to apply a specific region rather than to individual masses. This conversion is called Reynolds Transport Theorem and can be applied to all the basic laws. The next figure presents the system and control volume. At time t the volume is occupied by system is identical to the control volume. At time t t system moves to another location and the system and control volume possess different volumes. Now, consider an arbitrary flow field α ( x, y,z,t). We want to calculate the following integral: d α ( x, y,z,t ) dv dt V t s ( ) The above equation can be written in the following form: d 1 α dv = lim α ( t t ) dv α ( t ) dv dt V t 0 s ( t ) t Vs ( t t) Vs ( t ) 39 Reynolds Transport Theorem In the right hand side of the equation we add and subtract the following term: 1 t V ( t ) s α ( t ) t dv 40
Reynolds Transport Theorem The right hand side of the equation is as follows: 1 ( t t ) dv ( t t ) dv t α α Vs ( t t ) Vs ( t) 1 lim = α ( t t ) dv 1 t Vs ( t t ) Vs ( t ) α ( t t ) dv α ( t ) dv t Vs ( t ) Vs ( t ) 1 1 α t = t t dv dv s t Vs ( t t) Vs ( t ) V t C t 0 lim α ( t t ) α ( t ) dv α ( ) V t 0 ( t ) ( ) dv = n.u t da ( ) V ( t t ) V ( t ) A ( t ) s s s ( ) ( ) α t t dv = α t t n.u t da 41 Reynolds Transport Theorem Gauss-Ostrogradskii Divergence Theorem : If V is a closed region in space surrounded by a surface A, then: (. α u ) dv = α ( n. u ) V A da Using this theory the final form of the equation is as following: d α α ( t ) dv = ( u) dt. α dv Vs ( t ) V C t This equation is called Reynolds Transport Theorem. This equation states that the rate of increase of a m aterial quantity is equal to the rate of increase of that quantity in those particles inside fixed control volum e plus the net flux of the quantity through the boundaries of the control volum e. 42
Reynolds Transport Theorem The conservation equations can be derived using the R.T.T. by substitution of α with appropriate param eters as following. For m ass conservation equation α = ρ For m om entum conservation equation α = ρ u 1 2 For m om entum conservation equation α = ρ e u 2 43