CONVECTIVE HEAT TRANSFER

Mohammad GoharkhahDepartment of Mechanical Engineering, Sahand Unversity of Technology,

Tabriz, Iran

LAMINAR BOUNDARYLAYER FLOW

CHAPTER 3

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

LAMINAR BOUNDARYLAYER FLOW

LAMINAR BOUNDARYLAYER FLOW

Boundary layer theory was proposed byPrandtl shortly after the completion ofhis doctoral dissertation in 1904

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Introduction

TWO FUNDAMENTAL PROBLEMS IN CONVECTIVE HEAT TRANSFER1. The net force exerted by the stream on a plate (to calculate the pressure drop and

consequently the pumping power )

We must first determine the flow and temperature fields in the vicinity of the solid wall by solving the continuity, Navier-Stokes and energy equations

2. The resistance to the transfer of heat from theplate to the stream (to calculate the heat transferrate)

Key Question:How can we calculate F and q?

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

IntroductionThe conservation equations for an incompressible flow with constant property

Solve four equations with the boundary conditions to obtain four unknowns (u, v, P, T)

boundary conditions

The mathematical complexity of convection heat transfer is traced to the non-inearity of the Navier-Stokes equations of motion and the coupling of flow and thermal fields.

But!!!!

The boundary layer concept provides major simplifications.

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CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

The Boundary Layer Concept This concept is based on the notion that under special conditions certain terms in thegoverning equations are much smaller than others and therefore can be neglectedwithout significantly affecting the accuracy of the solution.Two questions are raised:(1) What are the conditions under which terms in the governing equations can bedropped?(2) What terms can be dropped?

velocity or viscous boundarylayer: Under certain conditions theeffect of viscosity is confined to a thinregion near the surface.

Thermal boundary layer: undercertain conditions the effect of thermalinteraction between the surface andthe moving fluid is confined to a thinregion near the surface.

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

The Boundary Layer ConceptThe velocity boundary layer (1)slender body without flow separation (2) high Reynolds number (Re >100). The thermal boundary layer : (1) Slender body without flow separation and (2) high product of Reynolds and Prandtl numbers (Pe=Re Pr > 100)

Important observations(1) Zero fluid velocity or so called no slip condition at the surface .(2) Fluid velocity and temperature change rapidly in the boundary layer. (Free stream

velocity and temperature at the edge of the boundary layer)(3) At high Re and Pr both velocity and thermal boundary layers are thin.(4) Viscosity plays no role outside the viscous boundary layer. Thus, the flow field isdevided into a viscosity dominated region (boundary layer), and an inviscid region(outside the boundary layer).(5) Boundary layers can exist in both forced and free convection flows.

The conditions for the formationof the two boundary layers.

Boundary Layer Equations- Scale analysis

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Boundary layer equations- Scale analysisThe scale of each of the five terms in the x-momentum equation is determined to see which term can be neglected

scales for changes in x, y, and u

Free stream characteristics

From mass continuity equation

The order of both inertia terms isThe inertia terms can not be neglected

∂2u/∂x2 term can be neglected at the expense of the ∂2u/∂y2 term

(slenderness of the boundary layer)

The next step is to simplify these two terms

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Boundary layer equations- Scale analysispressure ∼ friction balance Moreover, we know that :

(mass continuity)

inside the boundary layer, the pressure varieschiefly in the x direction;at any x, the pressure inside the boundary layerregion is practically the same as the pressureimmediately outside it,

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Boundary layer equations- Scale analysis

The boundary layer equation for energy

boundary layer equation for momentum

Three unknowns (u,v,T) are obtained from these equations. Compare this with the ‘‘four equations and four unknowns’’ problem contemplated originally.

The disappearance of the ∂2/∂x2 diffusion terms from the momentum and energy equations makes this new problem solvable in a variety of ways.

Now, we can write the boundary layer equations.

x∼L,

Performing a scale analysis to the energy equation we can simply show that the thermal diffusion in the x direction can be neglected.

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

SCALE ANALYSIS- boundary layer thickness and wall friction

SCALE ANALYSIS- boundary layer thickness and wall friction

(simplest free stream possible)Assume a free stream with uniform pressure

Consider the inertia ∼ friction balance in the boundary layer momentum equation

From mass continuity equation

Reynolds number based on the longitudinal dimension of the boundary layer region

slenderness postulate on which the boundary layer theory is based (δ <<L) is valid provided that ReL>>1

dimensionless skin friction coefficient

The real (measured or calculated) value of τ will differ from by only a factor of order unity

What is the order of magnitude of the boundary layer thickness and wall friction?The order of magnitude of the boundary layer thickness is obtained from momentum equation

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

SCALE ANALYSIS- Thermal boundary layer thickness and SCALE ANALYSIS- Thermal boundary layer thickness and Nusselt number

Scaling will now be used to find the order of magnitude of h and δT

x∼L, there is always a balance between conduction from the wall into the stream and convection

The δT scale needed for estimating h ∼ k/δT can be determined from the energy equation

Scales for u and v depend on whether δT is larger or smaller than δ

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Theaxial velocity u within the thermalboundary layer is smaller than the freestream velocity.Pretending that the velocity profile is linear,similarity of triangles gives a scale for u as

For this case the axial velocity u within the thermal boundary layer is of the order of the free stream velocity:

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

SCALE ANALYSIS- Thermal boundary layer thickness and SCALE ANALYSIS- Thermal boundary layer thickness and Nusselt number

Case1- Thick thermal boundary layer, δT >>δ

(mass continuity )The 2nd term can be neglected

the convection ∼ conduction balance becomes:Peclet number

the range occupied by liquid metalsδT >>δ,

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

SCALE ANALYSIS- Thermal boundary layer thickness and SCALE ANALYSIS- Thermal boundary layer thickness and Nusselt number

Case2- Thin thermal boundary layer, δT <<δ

fluids with Prandtl numbers of order 1 (e.g., air) or greater than 1 (e.g., water or oils).

These scaling results agree within a factor of order unity with the classical analytical results discussed next

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Comments

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Comments

δ is proportional to L1/2. Moreover, along the wall (0 < x < L), the boundary layerthickness increases as x 1/2. Now, one particular property of the x1/2 function is that itsslope is infinite at x = 0. This geometric feature of the boundary layer is inexplicablyabsent from the graphics employed by most texts.

The meaning of Reynolds number.:In most treatments of fluid mechanics, the Reynolds number is described as the order ofmagnitude of the inertia/friction ratio in a particular flow. This interpretation is notcorrect because in the boundary layer region, there is always a balance between inertiaand friction, whereas ReL can reach as high as 105 before the transition to turbulent flow.The only physical interpretation of the Reynolds number in boundary layer flow isgeometric

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Summary of Boundary Layer Equations for Steady Laminar Flow

Summary of Boundary Layer Equations for Steady Laminar Flow

Simplifying assumptions

ContinuumNewtonian fluid two-dimensional processnegligible changes in kinetic and potential energy constant propertiesSlender surfacehigh Reynolds number (Re > 100)high Peclet number (Pe > 100)steady statelaminar flowno dissipation no gravity no energy generation

Continuity:

x-Momentum:

Energy

The pressure term is obtained from the solution to inviscidflow outside the boundary layer. Thus , the momentum equation has two unknowns: u and v. To include the effect of buoyancy, the following term should be added to the right-hand-side of the momentum equation