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### Analysis of Convection Heat Transfer

### Computation of Dimensionless Temperature Profile

P M V Subbarao

Professor

Mechanical Engineering Department

IIT Delhi

Development of Design Rules …

Methods to evaluate convection heat transfer

- Empirical (experimental) analysis
- Use experimental measurements in a controlled lab setting to correlate heat and/or mass transfer in terms of the appropriate non-dimensional parameters
- Theoretical or Analytical approach
- Solving of the energy equations for a particular geometry.
- Example:
- Solve for q
- Use evaluate the local Nusselt number, Nux
- Compute local convection coefficient, hx
- Use these (integrate) to determine the average convection coefficient over the entire surface
- Exact solutions possible for simple cases.
- Approximate solutions are also possible using an integral method

Empirical method

- How to set up an experimental test?
- Let’s say you want to know the heat transfer rate of an airplane wing (with fuel inside) flying at steady conditions………….
- What are the parameters involved?
- Velocity, –wing length,
- Prandtl number, –viscosity,
- Nusselt number,
- Which of these can we control easily?
- Looking for the relation:Experience has shown the following relation works well:

insulation

Experimental test setup- Measure current (hence heat transfer) with various fluids and test conditions for
- Fluid properties are typically evaluated at the mean film temperature

Theoretical or Analytical approachSolving of the boundary layer equations for a particular geometry.

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

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.

T

T

T

T

T

T

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),
- Flat Plate Boundary Layer is an hypothetical standard for initiation of basic analysis.

Velocity Boundary Layer

d(x)

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 :

Momentum flux of far field stream:

The effect of solid boundary :

ratio of shear stress at wall/free stream Momentum flux

Thermal Boundary Layer

dT(x)

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 due to heat conduction. Thus, from the local heat flux:

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:

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

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

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

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

Other scales of reference:

Length of plate: L

Free stream velocity : u

First Law of Thermodynamics for A CV

Or Energy Equation for a CV

How to select A CV for External Flows ?

Conservation of Energy for A CV

- ECM= Energy of the system

The rate of change of energy of a control mass should be equal to difference of work and heat transfers.

Energy equation per unit volume:

Using the law of conduction heat transfer:

The net Rate of work done on the element is:

From Momentum equation: N S Equations

Substitute the work done by shear stress:

This is called the first law of thermodynamics for fluid motion.

Fis called as viscous dissipation.

Equation for Distribution of Temperature

The Boundary Layer : A Control Volume

Relative sizes of Momentum & Thermal Boundary Layers …

Boundary Layer Equations

Consider the flow over a parallel flat plate.

Assume two-dimensional, incompressible, steady flow with constant properties.

Neglect body forces and viscous dissipation.

The flow is nonreacting and there is no energy generation.

The governing equations for steady two dimensional incompressible fluid flow with negligible viscous dissipation:

Scale Analysis

Define characteristic parameters:

L : length

u∞: free stream velocity

T ∞: free stream temperature

x, y : positions (independent variables)

u, v : velocities (dependent variables)

T : temperature (dependent variable)

also, recall that momentum requires a pressure gradient for the movement of a fluid:

p : pressure (dependent variable)

Similarity parameters can be derived that relate one set of flow conditions to geometrically similar surfaces for a different set of flow conditions:

Boundary Layer Parameters

- Three main parameters (described below) that are used to characterize the size and shape of a boundary layer are:
- The boundary layer thickness,
- The displacement thickness, and
- The momentum thickness.
- Ratios of these thicknesses describe the shape of the boundary layer.

Because the boundary layer thickness is defined in terms of the velocity distribution, it is sometimes called the velocity thickness or the velocity boundary layer thickness.

- There are no general equations for boundary layer thickness.
- Specific equations exist for certain types of boundary layer.
- For a general boundary layer satisfying minimum boundary conditions:

The velocity profile that satisfies above conditions:

Laminar Velocity Boundary Layer

- The velocity boundary layer thickness for laminar flow over a flat plate:

- as u∞ increases, δ decreases (thinner boundary layer)

- The local friction coefficient:

- and the average friction coefficient over some distance x:

Laminar Thermal Boundary Layer: Blasius Similarity Solution

Boundary conditions:

This differential equation can be solved by numerical integration.

One important consequence of this solution is that, for pr >0.6:

Local convection heat transfer coefficient:

x

y

x

For large Pr (oils):

For small Pr (liquid metals):

Pr > 1000

Pr < 0.1

Fluid viscosity less than thermal diffusivity

Fluid viscosity greater than thermal diffusivity

A single correlation, which applies for all Prandtl numbers,

Has been developed by Churchill and Ozoe..

Transition to Turbulence

- When the boundary layer changes from a laminar flow to a turbulent flow it is referred to as transition.
- At a certain distance away from the leading edge, the flow begins to swirl and various layers of flow mix violently with each other.
- This violent mixing of the various layers, it signals that a transition from the smooth laminar flow near the edge to the turbulent flow away from the edge has occurred.

Flat Plate Boundary Layer Trasition

- Important point:
- Typically a turbulent boundary layer is preceded by a laminar boundary layer first upstream
- need to consider case with mixed boundary layer conditions!

Turbulent Flow Regime

- For a flat place boundary layer becomes turbulent at Rex ~ 5 X 105.
- The local friction coefficient is well correlated by an expression of the form

Local Nusselt number:

Mixed Boundary Layer

- In a flow past a long flat plate initially, the boundary layer will be laminar and then it will become turbulent.
- The distance at which this transitions starts is called critical distance (Xc) measured from edge and corresponding Reynolds number is called as Critical Reynolds number.
- If the length of the plate (L) is such that 0.95 Xc/L 1, the entire flow is approximated as laminar.
- When the transition occurs sufficiently upstream of the trailing edge, Xc/L 0.95, the surface average coefficients will be influenced by both laminar and turbulent boundary layers.

For a smooth flat plate: Rexc = 5 X 105

Array of Cylinders in Cross Flow

- The equivalent diameter is calculated as four times the net flow area as layout on the tube bank (for any pitch layout) divided by the wetted perimeter.

For triangular pitch:

Number of tube centre lines in a Shell:

Dsis the inner diameter of the shell.

Flow area associated with each tube bundle between baffles is:

where A s is the bundle cross flow area, Ds is the inner diameter of the shell, C is the clearance between adjacent tubes, and B is the baffle spacing.

the tube clearance C is expressed as:

Then the shell-side mass velocity is found with

Shell side Reynolds Number:

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