# Numerical Heat and Mass Transfer

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1 Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino

2 Introduction In conduction problems the convection entered the analysis merely as a boundary condition in the form of heat transfer coefficient. Our objective in this part of the course is to establish the physical and mathematical basis for the understanding of convective momentum and energy transport phenomena. In engineering applications, the pressure drop associated with flows inside ducts or over bodies is also of interest. The analysis of convection is complicated because the fluid motion affects the pressure drop, the drag force and the heat transfer. To determine the heat transfer coefficient, the velocity distribution in the flow is also needed, because the velocity enters the energy equation. The problem is governed by a set of coupled partial differential equations. Temperature field in the combustion chamber of a waste incinerator plant

3 Flow over a body ü When a fluid flows over a body, the velocity and temperature distribution at the immediate vicinity of the surface strongly influence the heat transfer by convection. ü The boundary-layer concept frequently is introduced to model the velocity and temperature fields near the solid surface in order to simplify the analysis of convective heat transfer. Velocity boundary layer To illustrate the concept of the velocity boundary layer, we consider the flow of a fluid over a flat plate. The fluid at the leading edge of the plate has a velocity u. As the fluid moves in the x-direction along the plate, the fluid particles that make contact with the plate surface assume zero velocity. The velocity boundary layer is the distance y=δ(x) from the surface where u=0.99 u. 3

4 Flow over a body With the boundary layer concept, the flow field can be separated in two distinct regions ü The boundary layer region, where the axial velocity component u(x,y) varies with the distance y from the plate and so the velocity gradients and the shear stress are considered large. ü The potential-flow region, where the velocity gradients and shear stresses are negligible. The characteristic of the flow is governed by the magnitude of the Reynolds number: Free-stream velocity Re u x ν Distance from leading edge Kinematic viscosity of fluid The boundary layer starts at the leading edge as a laminar boundary layer. Once a critical distance is reached (Re attains a critical value) fluid fluctuations begin to develop, which characterize the end of the laminar boundary layer and the beginning of transition to turbulent boundary layer. 4

5 Flow over a body Drag Coefficient and Drag Force Suppose the velocity profile u(x,y) in the boundary layer is known. The viscous shear stress τ x acting on the wall at any location x is determined from its definition by: Fluid viscosity τ x = µ u ( x,y) y y=0 Thus, knowing the velocity distribution in the boundary layer, one can determine the shear force acting on the wall owing to the flow. The above equation is not practical in engineering applications. In practice, the shear stress or the local drag force τ x per unit area is related to the local drag coefficient c x by the relation Fluid density Local drag coefficient ρu τ x = c x 5

6 Flow over a body The mean value of the drag coefficient over x=0 and x=l is defined as c m = 1 L Knowing the mean drag coefficient, we can find the drag force F acting on the plate form x=0 to x=l and for width w from L 0 c x dx ρu F = wlc m Thermal Boundary Layer Analogous to the concept of velocity boundary layer, one can envision the development of a thermal boundary layer along the flat plate associated with temperature profile in the fluid. To illustrate this concept, we consider that a fluid at a uniform temperature T flows along a flat plate maintained at a constant temperature T w. We define the dimensionless temperature T ( x,y) = T ( x,y ) T w T T w 6

7 Flow over a body At a distance sufficiently far from the wall, the fluid temperature remains the same as T. Therefore, at each location x along the plate, the thickness of the thermal boundary layer corresponds to a location y=δ t (x) in the fluid where the dimensionless temperature equals The relative thickness of the thermal boundary layer and the velocity boundary layer depends on the magnitude of the Prandtl number of the fluid. ü For fluids having a Pr number equal to unity, such as gases, δ(x)=δ t (x). ü The thermal boundary layer is much thicker than the velocity boundary layer for fluid having Pr<<1, such as liquid metals. ü The thermal boundary layer is much thinner than velocity boundary layer for fluids having Pr>>1. 7

8 Flow over a body Heat Transfer Coefficient Suppose the temperature distribution T(x,y) in the thermal boundary layer is known. Then the heat flux from the fluid to the wall is determined from: Fluid thermal conductivity T ( x,y) q ( x) = k y y=0 In engineering applications this relation is not practical to use to calculate heat transfer rate between the fluid and the wall. In practice, a local heat transfer coefficient h(x) is defined to calculate the heat flux between the fluid and the wall: q ( x) = h ( x) ( T T w ) h ( x) = k T T w T = k T y y=0 y y=0 Mean heat transfer Coefficient h m = 1 L L 0 h ( x )dx Heat transfer rate Q = h m A T T w 8

9 Flow over a body Relation between Drag Coefficient and Heat Transfer Coefficient If velocity profile and the boundary layer thickness are known, then an expression can be developed for the local drag coefficient c x for laminar flow along a flat plate. Similarly, if the temperature profile and the thermal boundary layer thickness are available, an expression can be developed for the local heat transfer coefficient for laminar flow along a plate. Are the heat transfer and drag coefficient related? The exact expressions for the local drag coefficient and the local Nusselt number for laminar flow along a flat plate are given by: c x = 0.33 Re x Nu x = 0.33 Pr 3 Re x Local Stanton number: St x = h ( x ) = ρ c p u h ( x) x k u x ν ν a = Nu x Pr Re x 9

10 Flow over a body Remembering the exact expression for local Nusselt number: Nu x 1 1 St x = 0.33 Pr 3 Re x Pr Re x = Pr Re x c x = Pr 3 c x St x Pr 3 = c x St m Pr 3 = c m ü This expression is referred as the Reynolds-Colburn analogy for laminar flow along a flat plate. ü It means that making a frictional drag measurement for laminar flow along a flat plate with no heat transfer involved, the corresponding heat transfer coefficient can be derived from the above equation. ü It is much easier to make drag measurement that heat transfer measurement! 10

11 Governing equations of fluid flow - introduction The governing equations of fluid flow represent mathematical statements of the conservation laws of physics: ü the mass of a fluid is conserved; ü the rate of change of momentum equals the sum of the forces on a fluid particle (Newton s second law); ü the rate of change of energy is equal to the sum of the rate of heat addition to and the rate of work done on a fluid particle (first law of thermodynamic). The fluid will be treated as a continuum and, for the analysis of fluid flows at macroscopic length scales (say 1mm and longer) the molecular structure of the matter and molecular motions may be ignored. The behaviour of the fluid is described in terms of macroscopic properties, such as velocity, pressure, density and temperature, and their space and time derivatives. This may be thought of as averages over suitably large number of molecules. 11

12 Governing equations of fluid flow - introduction We consider such a small element of fluid with sides δx, δy and δz. All fluid properties are functions of space and time so we would strictly need to write ρ(x,y,z,t), p(x,y,z,t) and u(x,y,z,t). The element under consideration is so small that the fluid properties at the faces can be expressed accurately enough by means of the first two terms of a Taylor series expansion. For instance, the pressure at the E and W faces are expressed by: p E = p + p x p W = p p x δx + o ( δx ) δx + o ( δx ) 1

13 Mass conservation equation Rate of increase of mass in fluid element = Net rate of flow of mass into fluid element ρ t δ x δ y δ z ρu δ = ρu x x δ δ ρu + ρu δ x y z x δ y δ z + ρv ( ρv ) δ y y δ δ ρv + ( ρv ) δ y x z y δ x δ z + ρw ( ρw ) δ z z δ δ ρw + ( ρw ) δ z x y z δ x δ y ρ t + div ( ρu) = 0 Compact vector notation div ( u) = u x + v y + w z = 0 Incompressible fluid 13

14 Rate of change following a fluid particle and for a fluid element The momentum and energy conservation equations make statements regarding the changes of properties of a fluid particle. Each property of such a particle if a function of the position (x,y,z) and time t. By referring to a generic fluid property ϕ per unit of mass, the time derivative of such a property with respect to time and following the particle is given by the substantive derivative: Dφ Dt = φ t + u φ x + v φ y + w φ z = φ t + u ( φ ) Per unit mass ü The substantive derivative defines the rate of change of property ϕ per unit mass. ü As in the case of mass conservation equation, we are interested in developing equations for rate of change per unit volume, given by the product of substantive derivative and density: ρ Dφ Dt = ρ φ t + u φ Per unit volume 14

15 Rate of change following a fluid particle and for a fluid element The substantive derivative follows the fluid particle. In the following the relationship between the substantive derivative of ϕ and the rate of change of ϕ for a fluid element is underlined. The generalization of the mass conservation equation for an arbitrary conserved property of f is: ( ρφ ) + div ( ρφu ) = ρ φ t t + u φ + φ ρ + div ( ρu) t Mass conservation = ρ Dφ Dt Rate of increase of ϕ of fluid element Net rate of flow of ϕ out of fluid element Rate of increase of ϕ for a fluid particle Both the conservative (or divergence) form and non-conservative form of the rate of change can be used as alternatives to express the conservation of physical quantity. The non conservative forms are usually used in the derivation of the momentum and energy conservation equation for brevity of notation and to emphasize that the conservation laws are fundamentally conceived as statements that apply to fluid particle. 15

16 Momentum conservation equation Newton s second law states that the rate of change of momentum of a fluid particle equals the sum of the forces on the particles. ρ Du Dt = F Surface Forces Body forces x) Pressure forces Viscous forces ( p + τ xx ) + τ yx x y + τ zx z Gravity forces Centrifugal forces Coriolis force Electromagnetic forces y) z) ( p + τ yy ) + τ xy y x + τ zy z ( p + τ zz ) + τ xz z x + τ yz y Stress component in the x-direction 16

17 Momentum conservation equation The components of the momentum conservation equations are found by setting the rate of change of fluid particle equal to the total force in the considered direction. x) ρ Du Dt = p + τ xx x y) ρ Dv Dt = p + τ yy y z) ρ Dw Dt = p + τ zz z Navier-Stokes equations for a Newtonian Fluid + τ yx y + τ zx z + S Mx + τ xy x + τ zy z + S My + τ xz x + τ yz y + S My x-, y and z-component of the body force ü The most useful of momentum conservation equations are obtained by introducing a suitable model for viscous stresses. ü In many fluid flows the viscous stresses can be expressed as a function of the local deformation rate. ü In three-dimensional flows the local rate of deformation is composed of the linear deformation rate and the volumetric deformation rate. 17

18 Momentum conservation equation ε xx = u x ε yy = v y ε zz = w z ε xy = ε yx = 1 u y + v x ε xz = ε zx = 1 u z + w x ε yz = ε zy = 1 v z + w y Linear deformation rates u v w + + = div x y z ( u) Volumetric deformation rate In a Newtonian fluid the viscous stresses are proportional to the rates of deformation. ü Linear deformations and stresses are related by dynamic viscosity μ. ü Volumetric deformation and stresses are related by viscosity λ. τ xx = µ u x + λdiv u τ yy = µ v y + λdiv u τ zz = µ w z + λdiv u τ xy = τ yx = µ u y + v x τ xz = τ zx = µ u z + w x τ yz = τ zy = µ v z + w y Substitution of these shear stresses into the momentum conservation equations yields the so-called Navier- Stokes equations, named after the two 19 th century scientists who derived them independently 18

19 Energy conservation equation The statement of the energy conservation equation for a fluid particle is: ρ De Dt =Q L where e = i + u + v + w + gz Gravitational force can be regarded as a body force which does work on the fluid element. This force is represented by term S E in the governing equation. Work done by the surface forces: x) y) z) u p + τ xx x v ( p + τ yy ) y w ( p + τ zz ) z + ( uτ yx ) y + ( vτ xy ) x + ( wτ xz ) x + uτ zx δ z x δ y δ z + vτ zy δ z x δ y δ z + wτ yz δ y x δ y δ z 19

20 Energy conservation equation Energy flux by conduction The heat flux vector has three components. The net rate of heat transfer to the fluid particle due to heat flow in the x-direction is given by: q x q x x 1 δ x q + q x x x 1 δ x δ y δ z = q x x δ x δ y δ z The total rate of heat added to the fluid particle per unit volume due to heat flow across its boundaries is the sum of net heat transfer rates in the three directions divided by the volume: q x x q y y q z z = div q q = k T q = k ( T ) 0

21 Energy conservation equation The energy equation is then: e = i + u + v + w h = u + p ρ ρ De Dt = ( pu ) + ( uτ xy ) y + uτ xx x + ( vτ yx ) + vτ yy x y + ( wτ zx ) + wτ zy x y + k T + S E + uτ xz z + ( vτ yz ) z + ( wτ zz ) z τ xx = µ u x + λdiv ( u ) τ yy = µ v y + λdiv ( u ) τ zz = µ w z + λdiv ( u ) τ xy = τ yx = µ u y + v x τ xz = τ zx = µ u z + w x τ yz = τ zy = µ v z + w y 1

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