CONVECTION : An Activity at Solid Boundary

CONVECTION : An Activity at Solid Boundary

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Identify and Compute Gradients at Boundary …..

Heat Transfer in Equilibrium Layer

• The thickness of stagnant layer decides the magnitude of normal temperature gradient at the wall.

• And hence, the thickness of wall fluid layer decides the magnitude of convective heat transfer coefficient.

• Typically, the convective heat transfer coefficient for laminar flow is relatively low compared to the convective heat transfer coefficient for turbulent flow.

• This is due to turbulent flow having a thinner stagnant fluid film layer on the heat transfer surface.

At the wall for fluid layer :

TThAy

TAk wallfluidfluid ,

TT

yT

k

hwall

layerwall

fluid

At Thermodynamic equilibrium

wallwallfluid TT ,

Estimation of Heat Transfer Coefficient

• Estimation of heat transfer coefficient is basically computation of temperature profile.

• A general theoretical and experimental study to understand how the stagnant layer is developed.

• The global geometry of the solid wall and flow conditions will decide the structure of stagnant layer.

• Basic Geometry : Internal Flow or External Flow.

TT

yT

k

hwall

wall

fluid

Internal Flows

• Internal flow can be described as a flow whose boundary layer is eventually constrained as it develops along an adjacent surface.

• The objectives are to determine if:• the flow is fully developed (no variation in the

direction of the flow• laminar or turbulent conditions• the heat transfer

Entrance and developed flows

q’’

Ti

Ts(x)

Ti Ts(x)q’’

Hot Wall & Cold Fluid

Cold Wall & Hot Fluid

Temperature Profile in Internal Flow

External Flows

• Any property of flow can have a maximum difference of Solid and free stream properties.

• There will be continuous growth of Solid surface affected region in Main stream direction.

• The extent of this region is very very small when compared to the entire flow domain.

• Free stream flow and thermal properties exit during the entire flow.

A continuously Growing Solid affected Region.

The Boundary Layer

CONVECTION BOUNDARY LAYER

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

A tiny but very effective part of A Fluid Flow……

De Alembert to Prandtl

17521904

Ideal to Real1822

1860

Introduction

• A boundary layer is a thin region in the fluid adjacent to a surface where velocity, temperature and/or concentration gradients normal to the surface are significant.

• Typically, the flow is predominantly in one direction.• As the fluid moves over a surface, a velocity gradient is

present in a region known as the velocity boundary layer, δ(x).

• Likewise, a temperature gradient forms (T ∞ ≠ Ts) in the thermal boundary layer, δt(x),

• Therefore, examine the boundary layer at the surface (y = 0).

• Flat Plate Boundary Layer is an hypothetical standard for initiation of basic analysis.

Velocity Boundary Layer

Fluid particles in contact with the surface have zero velocity

u(y=0) = 0; no-slip boundary condition

Fluid particles in adjoining layers are retarded

δ(x): velocity boundary layer thickness

At the surface there is no relative motion between fluid and solid.

The local momentum flux (gain or loss) is defied by Newton’s Law of Viscosity :

0

y

wall y

u

Momentum flux of far field stream:

2

2''

uP

The effect of solid boundary : ratio of shear stress at wall/free stream Momentum flux

Coefficient of friction:

2

2

0

''

u

yu

PC ywall

f

Thermal Boundary Layer

Fluid particles in contact with the surface attain thermal equilibrium

T(y=0) = Ts

Fluid particles transfer energy to adjoining layers

δt (x): thermal boundary layer thickness

Hot Surface Thermal Boundary Layer

Plate surface is warmer than the fluid (Ts > T∞)

Cold Surface Thermal Boundary Layer

Plate surface is cooler than the fluid (Ts < T∞)

At the surface, there is no fluid motion, heat transfer is only possible dueto heat conduction. Thus, from the local heat flux:

0

''

y

y

Tkq

wall

This is the basic mechanism for heat transfer from solid to liquid or Vice versa.

The heat conducted into the fluid will further propagate into free stream fluid by convection alone.

Use of Newton’s Law of Cooling:

TThq s''

0

''

yys y

TkTThq

Temperature distribution in a boundary layer of a fluid depends on:

pL

s

ckdx

dpxf

TT

TT,:,Re,

*

**

Scale of temperature:

ss TTTT

:,Re,*

**

dz

dpxf

TT

TTL

s

Pr,,Re,

*

**

dx

dpxf L

Prandtl Number: The ratio of momentum diffusion to heat diffusion.

T

m

Pr

Other scales of reference:

Length of plate: L

Free stream velocity : uoo

Potential for diffusion of momentum change (Deficit or excess) created by a solid boundary.

Potential for Diffusion of thermal changes created by a solid boundary.

0

''

y

s y

TkTThq

0*

0 *scaleLength

scale eTemperatur

yyyy

T

0

**

y

sfluids yL

TTkTTh

Pr,,Re,*

**

0*

* dx

dpxf

k

hL

y Lfluidy

This dimensionless temperature gradient at the wall is named asNusselt Number:

resistance Convection

resistance Conduction1

h

kL

k

hLNu fluid

fluid

Pr,,Re,*

**

0*

* dx

dpxf

k

hL

yNu L

fluidy

Local Nusselt Number

Average Nusselt Number

avgfluid

avgavg k

LhNu

,

Computation of Dimensionless Temperature Profile

First Law of Thermodynamics for A CV

Energy Equation for a CV

How to select A CV for External Flows ?

Relative sizes of Momentum & Thermal Boundary Layers …

T

m

Pr