Heat transfer Hw 4 Chapter 6

Chapter-6IntroductiontoConvection.pdf

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Fundamentals of Heat and Mass Transfer, Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera, David P. DeWitt, John Wiley & Sons, Inc.

•Chapter 1: Introduction

Conduction Heat Transfer •Chapter 2: Introduction to Conduction •Chapter 3: 1D, Steady-State Conduction •Chapter 4: 2D, Steady-State Conduction •Chapter 5: Transient Conduction

Convection Heat Transfer •Chapter 6: Introduction to Convection •Chapter 7: External Flow •Chapter 8: Internal Flow •Chapter 9: Free Convection •Chapter 10: Boiling and Condensation •Chapter 11: Heat Exchangers

Radiation Heat Transfer •Chapter 12: Radiation Processes and Properties •Chapter 13: Radiation Exchange Between Surfaces

1 Mass Transfer

•Chapter 14: Diffusion Mass Transfer

Chapter-6

(Introduction to Convection) 2

Chapter-6: Introduction to Convection (1/2)

6.1 The Convection Boundary Layers 378 6.1.1 The Velocity Boundary Layer 6.1.2 The Thermal Boundary Layer 6.1.3 The Concentration Boundary Layer 6.1.4 Significance of the Boundary Layers

6.2 Local and Average Convection Coefficients 6.2.1 Heat Transfer 6.2.2 Mass Transfer 6.2.3 The Problem of Convection

6.3 Laminar and Turbulent Flow 6.3.1 Laminar and Turbulent Velocity Boundary Layers 6.3.2 Laminar and Turbulent Thermal and Species Concentration Boundary Layers

Chapter-6: Introduction to Convection (2/2)

6.4 The Boundary Layer Equations 6.4.1 Boundary Layer Equations for Laminar Flow 6.4.2 Compressible Flow

6.5 Boundary Layer Similarity: The Normalized Boundary Layer Equations 6.5.1 Boundary Layer Similarity Parameters 6.5.2 Functional Form of the Solutions

6.6 Physical Interpretation of the Dimensionless Parameters 6.7 Boundary Layer Analogies

6.7.1 The Heat and Mass Transfer Analogy 6.7.2 Evaporative Cooling 6.7.3 The Reynolds Analogy

6.8 Summary

Boundary Layers: Physical Features

• Velocity Boundary Layer – A consequence of viscous effects

associated with relative motion between a fluid and a surface.

– A region of the flow characterized by shear stresses and velocity gradients.

– A region between the surface and the free stream whose thickness ( ) increases in the flow direction.

– Why does increase in the flow direction?

– Manifested by a surface shear

stress that provides a drag force (FD).

– How does vary in the flow direction? Why?

Thermal Boundary Layer

– A consequence of heat transfer between the surface and fluid.

– A region of the flow characterized by temperature gradients and heat fluxes.

– A region between the surface and the free stream whose thickness increases in the flow direction.

– Why does increase in the flow direction?

– Manifested by a surface heat flux qsʹʹ and a convection heat transfer coefficient h.

– If is constant, how doand h vary in the flow direction?

Distinction between Local and Average Heat Transfer Coefficients

• Local Heat Flux and Coefficient:

• Average Heat Flux and Coefficient for a Uniform Surface

Temperature: q = ∫As q ʹʹ

dAs

• For a flat plate in parallel flow:

Boundary Layer Transition

• How would you characterize conditions in the laminar region of boundary layer development? In the turbulent region? • What conditions are associated with transition from laminar to turbulent flow? • Why is the Reynolds number an appropriate parameter for quantifying transition from laminar to turbulent flow? • Transition criterion for a flat plate in parallel flow:

What may be said about transition if ReL<Rex,c? If ReL>Rex,c?

• Effect of transition on boundary layer thickness and local convection coefficient:

Why does transition provide a significant increase in the boundary layer thickness?

Why does the convection coefficient decay in the laminar region? Why does it increase significantly with transition to turbulence, despite the increase in the boundary layer thickness? Why does the convection coefficient decay in the turbulent region?

Boundary Layer Equations (1/3)

• Consider concurrent velocity and thermal boundary layer development for steady, two-dimensional, incompressible flow with constant fluid properties and negligible body forces.

• Apply conservation of mass, Newton’s 2nd Law of Motion and conservation of energy

to a differential control volume and invoke the boundary layer approximations.

Velocity Boundary Layer:

Thermal Boundary Layer:

Boundary Layer Equations (2/3)

Conservation of Mass:

In the context of flow through a differential control volume, what is the physical significance of the foregoing terms, if each is multiplied by the mass density of the fluid?

• Newton’s Second Law of Motion:

What is the physical significance of each term in the foregoing equation? Why can we express the pressure gradient as dp∞/dx instead of ∂p / ∂x ?

Boundary Layer Equations (3/3)

Conservation of Energy:

What is the physical significance of each term in the foregoing equation?

What is the second term on the right-hand side called and under what conditions may it be neglected?

Boundary Layer Similarity (1/4)

• As applied to the boundary layers, the principle of similarity is based on determining similarity parameters that facilitate application of results obtained for a surface experiencing one set of conditions to geometrically similar surfaces experiencing different conditions. (Recall how introduction of the similarity parameters Bi and Fo permitted generalization of results for transient, one-dimensional conduction). • Dependent boundary layer variables of interest are: • For a prescribed geometry, the corresponding independent variables are:

Geometrical: Size (L), Location (x,y)

Hydrodynamic: Velocity (V)

Fluid Properties:

and

T = f (x , y , L, V , ρ , µ, c p , k , Ts ,T∞ ) Boundaryh=f(x,L,V,ρ,µ, c Layerp,k,Ts,T∞) Similarity (2/4)

• Key similarity parameters may be inferred by non-dimensionalizing the momentum and energy equations.

• Recast the boundary layer equations by introducing dimensionless forms of the independent and dependent variables. • Neglecting viscous dissipation, the following normalized forms of the x-momentum

and energy equations are obtained:

Boundary Layer Similarity (3/4)

How may the Reynolds and Prandtl numbers be interpreted physically? • For a prescribed geometry,

The dimensionless shear stress, or local friction coefficient, is then

What is the functional dependence of the average friction coefficient?

Boundary Layer Similarity (4/4)

For a prescribed geometry,

The dimensionless local convection coefficient is then

What is the functional dependence of the average Nusselt number?

How does the Nusselt number differ from the Biot number?

Reynolds Analogy (1/2)

• Equivalence of dimensionless momentum and energy equations for negligible pressure gradient (dp*/dx*~0) and Pr~1:

Advection terms Diffusion

• Hence, for equivalent boundary conditions, the solutions are of the same form:

u * = T * ∂u * = ∂T

∂y * ∂y* * = 0

* =0 y y

C f Re

2 = Nu

Reynolds Analogy (2/2)

With Pr = 1, the Reynolds analogy, which relates important parameters of the

velocity and thermal boundary layers, is • Modified Reynolds (Chilton-Colburn) Analogy: – An empirical result that extends applicability of the Reynolds analogy:

Colburn j factor for heat transfer

– Applicable to laminar flow if dp*/dx* ~ 0.

– Generally applicable to turbulent flow without restriction on dp*/dx*.

Exercise Problem 6.28: Determination of heat transfer rate for prescribed turbine blade operating conditions from wind tunnel data obtained for a geometrically similar but smaller blade. The blade surface area may be assumed to be directly proportional to its characteristic length . (1/2)

SCHEMATIC:

Exercise Problem 6.39: Use of a local Nusselt number correlation to estimate the surface temperature of a chip on a circuit board. (1/3)

SCHEMATIC:

Exercise Problem 6.39: Use of a local Nusselt number correlation to estimate the surface temperature of a chip on a circuit board. (2/3)

Exercise Problem 6.39: Use of a local Nusselt number correlation to estimate the surface temperature of a chip on a circuit board. (3/3)

Concentration Boundary Layer

• Features – A consequence of evaporation or sublimation of species A from a liquid or solid surface across which a second fluid species B is flowing. – A region of the flow characterized by species fluxes and concentration gradients.

– A region between the surface and free stream whose thickness increases in the flow direction.

– Why does increase in the flow direction? – Manifested by a surface species flux and a convection mass transfer coefficient hm. – What is the heat transfer analog to Fick’s law?

δ c → CA, s − CA ( y ) = 0.99

C A, s −

A,∞

Definitions (1/2)

Term Variable Units

Species molar flux kmol/s·m 2

Species molar rate kmol/s

Species mass flux kg/s·m 2

Species mass rate kg/s

Species molar concentration kmol/m 3

Species mass concentration (density) kg/m 3

Species molecular weight kg/kmol

Convection mass transfer coefficient m/s

Binary diffusion coefficient1 m 2/s

1 Table A.8

Definitions (2/2)

• Convection Calculations

Species Molar Flux:

Species Mass Flux:

Total Transfer Rates:

Average Mass Transfer Coefficient:

Species Vapor Concentration or Density (1/2)

• At a Vapor/Liquid or Vapor/Solid Interface

The vapor concentration/density corresponds to saturated conditions at the interface temperature Ts .

Assuming perfect gas behavior, the concentration/density may be estimated from knowledge of the saturation pressure.

The concentration may also be directly determined from saturation tables. For example, from Table A.6 for saturated water,

Species Vapor Concentration or Density (2/2)

• Free Stream Conditions

– The free stream concentration/density may be determined from knowledge of the vapor pressure, assuming perfect gas behavior.

– For water vapor in air, the free stream concentration/density may be determined from knowledge of the relative humidity, .

For dry air,

Species Boundary Layer Equation and Similarity (1/3)

• Species boundary layer approximation: • Species equation for a non-reacting

boundary layer:

What is the physical significance of each term?

Is this equation analogous to another boundary layer equation?

Species Boundary− Layer C* ≡ CA CA,s

EquationACA,∞− CandA,s Similarity (2/3)

• Dimensionless form of the species boundary layer equation:

How may the Schmidt number be interpreted? • Functional dependence for a prescribed geometry:

Species Boundary Layer Equation and Similarity (3/3)

The dimensionless local convection mass transfer coefficient is then

What is the functional dependence of the average Sherwood number?

Analogies (1/2)

• Heat and Mass Transfer Analogy From analogous forms of the dimensionless boundary layer energy and species equations, it follows that, for a prescribed geometry and equivalent boundary conditions, the functional dependencies of Nu and Sh are equivalent.

Since the Pr and Sc dependence of Nu and Sh, respectively, is typically of the form Prn and Scn, where n is a positive exponent (0.30 ≤ n ≤ 0.40),

Analogies (2/2)

• Reynolds Analogy

• Modified Reynolds Analogy

Colburn j factor for mass transfer

– Applicable to laminar flow if dp*/dx* ~ 0.

– Generally applicable to turbulent flow without restriction on dp*/dx*.

Evaporative Cooling (1/2)

• The term evaporative cooling originates from association of the latent energy created by evaporation at a liquid interface with a reduction in the thermal energy of the liquid. If evaporation occurs in the absence of other energy transfer processes, the thermal energy, and hence the temperature of the liquid, must decrease. • If the liquid is to be maintained at a fixed temperature, energy loss due to evaporation must be replenished by other means. Assuming convection heat transfer at the interface to provide the only means of energy inflow to the liquid, an energy balance yields

Evaporative Cooling (2/2)

Steady-state

Obtained from heat/mass

Cooling transfer analogy

• With radiation from the interface and heat addition by other means,

Exercise Problem 6.64: Use of the naphthalene sublimation technique to obtain the average convection heat transfer coefficient for a gas turbine blade. (1/3)

Schematic:

Exercise Problem 6.64: Use of the naphthalene sublimation technique to obtain the average convection heat transfer coefficient for a gas turbine blade. (2/3)

Exercise Problem 6.64: Use of the naphthalene sublimation technique to obtain the average convection heat transfer coefficient for a gas turbine blade. (3/3)

Exercise Problem 6.80: Use of wet and dry bulb temperature measurements to determine the relative humidity of an air stream. (1/2)

SCHEMATIC:

(318K): vg = 15.52 m 3/kg; Table A-8, Air-vapor (1 atm, 298 K): D AB= 0.26 ⋅ 10

-4m2/s,

Exercise Problem 6.80: Use of wet and dry bulb temperature measurements to determine the relative humidity of an air stream. (2/2)