HW #4 /Tutorial # 4 WRF Chapter 18; WWWR Chapter 19 ID Chapter 6courses.nus.edu.sg/course/chewch/CN2125/lectures/Week4.pdf · 2019. 12. 16. · HW #4 /Tutorial # 4 WRF Chapter 18;

HW #4 /Tutorial # 4

WRF Chapter 18; WWWR Chapter 19

ID Chapter 6

• Tutorial # 4

• WWWR #19.1,19.4, WRF#18.10.

• ID # 6.33. 6.37.

• To be discussed during the week 8-12 Feb., 2021.

• By either volunteer or class list.

• Homework # 4 (Self practice)

• WRF#18.1, 18.3 ID # 6.35, 6.36, 6.39

# Correction

• Question 19.19

1/ 2 1/ 4 1/ 4

1/ 2 1/ 4 1/ 4

3.94Pr (Pr 0.954)

0.508Pr (Pr 0.954) ;

x

x x x x

Grx

kNu Gr h Nu

x

Convective Heat Transfer

Fundamental Considerations In

Convective Heat Transfer

• Two main classifications of convective heat

transfer

• These have to do with the driving force causing

fluid to flow

Natural or free convection

Fluid motion results from heat transfer

Fluid heated/ cooled -> density change/ buoyancy effect ->

natural circulation in which affected fluid moves of its own

accord past the solid surface

- fluid it replaces is similarly affected by the energy transfer -

process is repeated

Forced convection

Fluid circulation is produced by external agency (fan or pump)

Analytical Methods

(a) Dimensional Analysis

(b) Exact Analysis of the Boundary Layer

(c) Approximate Integral Analysis

(d) Analogy between Energy and Momentum Exchange

Significant Parameters In

Convective Heat Transfer

A. Both have same dimensions L2/t; thus their ratio must be

dimensionless

B. This ratio, that of molecular diffusivity of momentum to the

molecular diffusivity of heat, is designed the Prandtl number

Pr =n

a

mcpk

Prandtl number

•observed to be a combination of fluid properties;

•thus Pr itself may be thought of as a property.

•Primarily a function of temperature

s

A ratio of conductive thermal resistance to the convective thermal

resistance of the fluid

Nusselt numberNu hL

k

Where the thermal conductivity of the fluid as opposed to that of the

solid, which was the case in the evaluation of the Biot modulus.

Dimensional Analysis of

Convective Energy Transfer

Forced Convection

Dimensional

Analysis for

Forced Convection

Natural Convection

Dimensional

Analysis for

Natural Convection

Courtesy Contribution

by ChBE Year

Representative, 2004.

Exact Analysis of the Laminar

Boundary Layer

A logical consequence of this situation is that the hydrodynamic

and thermal boundary layers are of equal thickness.

It is significant that the Prandtl numbers for most gases are

sufficiently closed to unity that the hydrodynamic and thermal

boundaries are of similar extent.

See detailed derivation

below

Detailed Derivation

Eq.(19-19) WWWR page 295

0

0

2

Re0.664

2

y

S

x

S

T T

y y

T T f

y

T TX

Re2

x

y

x

From Eq.(19-18)

Re

0.332x

ST TX

1

20.332

ReS xT TX

(19-19)

Approximate Integral Analysis of

the Thermal Boundary Layer

An approximate method for analysis of the thermal boundary layer

employs the integral analysis used by von Kármán for the

hydrodynamic boundary layer.

Consider the control volume designated by the dashed lines,

applying to flow parallel to a flat surface with no pressure

gradient, having width Dx, a height equal to the thickness of the

thermal boundary layer

0.323

Energy and Momentum Transfer

Analogies

The Colburn analogy expression is

St Pr 2/3 = Cf2

(19-37)

8)

9)

s

oner

Example 1

Water at 50o

F enters a heat-exchanger tube having an inside diameter of 1 in and a length of 10

ft. The water flows at 20 gal/min. For a constant wall temperature of 210o

F estimate the

Film temperature = (water mean bulk temperature + pipe wall temperature)/2

Mean bulk temperature of water = (inlet + outlet)/2

=(90+210)/2 = 150 = (50+130)/2

Second iteration is required since if |TL – 130| >

3o

F?