HMTchap7

Chapter 7

External Forced Convection

In this course, discussion is limited to low speed, forced convection,

with no phase change within the fluid.

7.1 The Empirical Method

The empirical correlation of the form

may be determined experimentally, as shown in Figs. 7.1-7.2.

(6.49)

(6.50)

(7.1)

),*,( PrRexfNu xx

),( PrRefNu xx

nmLL PrCReNu

for fixed

log log log

for fixed

log log log

L L

L

L

Pr

Nu C m Re

Re

Nu C n Pr

Note: To account for the effect of non-uniform

temperature on fluid properties, there are two methods:

(1) To evaluate fluid properties at the film temperature

(2) To evaluate all properties at and to correct with a

parameter of the form (Pr/Prs)r or (m/ms)

r.

nmLL PrCReNu

T f Ts T

2

(7.1)

m nL LSh CRe Sc (7.3)

(7.2)

Heat transfer:

Mass transfer:

7.2 The Flat Plate in Parallel Flow 7.2.1 Laminar Flow Over an Isothermal Plate: A Similarity Solution

Read the textbook pp. 437-443.

With stream function y(x, y) and the transformation of h, f (h) defined in (7.9), (7.10), the continuity equation (7.4) together with

the steady-state momentum equation (7.5) can be reduced to a

single ordinary differential equation (7.17), subjected to BCs

(7.18). The numerical solution is shown in Table 7.1. The

boundary layer thickness d and the local friction coefficient Cf,x can be determined as (7.19) and (7.20).

3 2

3 2

0

2 0, (0) 0; 1d f d f df df

f fd d d d

h hh h h h

; //

f y u xu x u

yh

(7.17,18)

(7.21,22)

2

2

* *0, *(0) 0; *( ) 1

2

d T Pr dTf T T

d dh h

(7.9,10)

7.2.1 Laminar Flow Over an Isothermal Plate: A Similarity Solution

Similarly, the energy equation can be reduced to (7.21), subjected to BCs (7.22). Numerical integration leads to

and

From the solution of (7.21), it also follows that

The average heat transfer coefficient is

Hence,

Similarly,

For small Pr, namely liquid metals, dt >>d, we may assume u=u throughout the thermal boundary layer and obtain (7.32).

For all Pr numbers: (7.33)

3/1

0

332.0*

Prd

dT

hh

1/ 2 1/30.332 0.6xx xh x

Nu Re Pr Prk

3/1Prt

d

d

xx

xx hdxhx

h 21

0

1/ 2 1/30.664 0.6xx xh x

Nu Re Pr Prk

, 1/ 2 1/3

AB

0.664 0.6m x

x x

h xSh Re Sc Sc

D

(7.23)

(7.24)

(7.30)

(7.31)

, 1/ 2

, 20.664

/ 2

s x

f x xC Reu

(7.20)

7.2.2 Turbulent Flow over an Isothermal Plate

From experiment, it is known

Moreover,

For turbulent flow,

Using (7.35) with the modified Reynolds analogy,

Enhanced mixing causes the turbulent boundary layer to grow more rapidly than the laminar boundary layer (d varies as x4/5 in contrast to x1/2 for laminar flow) and to have larger friction and convection coefficients.

1/5 7

, 0.0592 10f x x xC Re Re

5/137.0 xxRed

d d t dc

4/5 1/30.0296 0.6<

7.2.3 Mixed boundary Layer Conditions

When transition occurs sufficiently upstream of the trailing edge,

(xc/L)0.95 (Fig. 7.3), both the laminar and the boundary layers

should be considered.

General expressions:

1/5 8,

20.074 10f L L x,c L

L

AC Re Re Re

Re

4/5 1/3

8

0.6 60(0.037 )

10L L

x,c L

PrNu Re A Pr

Re Re

4/5 1/3

8

0.6 60(0.037 )

10L L

x,c L

ScSh Re A Sc

Re Re

(7.38)

(7.40)

(7.41)

4/5 1/ 2

, ,0.037 0.664x c x cA Re Re (7.39)

7.2.4 Unheated Starting Length

For flat plate in parallel flow with unheated starting length (Fig. 7.5):

Laminar flow:

turbulent flow:

0

1/33/ 41 ( / )

x

x

NuNu

x

0

1/99/101 ( / )

x

x

NuNu

x

(7.42)

(7.43)

7.5

7.2.5 Flat Plates with Constant Heat Flux Conditions

For flat plate with a uniform surface heat flux:

laminar flow: (7.45)

turbulent flow: (7.46)

In this case, is varying. The local surface temperature is

Note: Any of the results obtained for a uniform surface

temperature may be used with Eq. 7.48 to evaluate .

"

( ) ssx

qT x T

h

"

0 0

"

1/ 2 1/3

1 ( ) ( )

where 0.680

L Ls

s s

x

s

L

L L

q xT T T T dx dx

L L kNu

q L

k Nu

Nu Re Pr

1/ 2 1/30.453 , 0.6xx xh x

Nu Re Pr Prk

in comparison with (7.23) 4/5 1/30.0308 , 0.6 60x x xNu StRe Pr Re Pr Pr

in comparison with (7.36)

(7.48)

(7.49)

LNu( )sT T

7.2.6 Limitations on Use of Convection Coefficients

Errors as large as 25% may be incurred by using the expressions

due to varying free stream turbulence and surface roughness.

7.3 Methodology for a Convection Calculation

Follow the six steps listed in the textbook.

1. Become immediately cognizant of the flow geometry.

2. Specify the appropriate reference temperature and evaluate the

pertinent fluid properties at that temperature.

3. In mass transfer problems the pertinent fluid properties are those

of species B.

4. Calculate the Reynolds number.

5. Decide whether a local or surface average coefficient is required.

6. Select the appropriate correlation.

EXs 7.1-7.3

7.4 The Cylinder in Cross Flow

Boundary layer separation may occur due to the adverse pressure

gradient (dp/dx > 0). Boundary layer transition, from laminar b. l.

to turbulent b. l., depends on ReD (VD/m).

Drag coeff. CDFig. 7.9

7.6

7.8

7.7

7.9

7.4.2 Convection Heat and Mass Transfer

The local Nuq Fig. 7.10

Rise due to mixing

in the wake

Decline due boundary

layer growth Rise due to

transition to

turbulence

7.10

The average (of more engineering interest):

Constants for circular cylinders: Table 7.2

1/3mD D

hDNu CRe Pr

k --empirical (7.52)

7.52

Constants for noncircular cylinders: Table 7.3

1/3mD D

hDNu CRe Pr

k (7.52)

Other correlations are shown in (7.53), (7.54).

EX 7.4

52

7.5 The Sphere

Whitaker:

For liquid drops the Ranz and Marshall correlation (7.57). More accurate modifications for it are also available.

EX 7.6

1/4

1/ 2 2/3 0.4

s

4

s

2 (0.4 0.06 )

0.71 380

3.5 7.6 10

1.0 ( / ) 3.2

D D D

D

Nu Re Re Pr

Pr

Re

m

m

m m

(7.56)

1/ 2 1/32 0.6D DNu Re Pr (7.57)

7.6 Flow across Banks of Tubes (brief introduction)

7.11

7.12

Aligned or staggered: Fig. 7.12 and Fig. 7.13

A number of correlations for are given in (7.58)-(7.65), with constants listed in Tables 7.5-7.8.

The h for a tube in the first row is approximately equal to that for a single tube in cross flow, whereas larger heat transfer coefficients are associated with tubes of the inner rows. Mostly, the convection coefficient stabilizes for a tube beyond the fourth or fifth row.

In general, heat transfer enhancement is favored by the more tortuous flow of a staggered arrangement, particularly for small ReD (

7.7 Impinging Jets (brief introduction)

The impinging jet is a simple and effective way of cooling. The

compressed boundary layer near the central stagnation point

causes effective cooling of the surface. It is widely adopted, in

combination of cooling fins, in CPU cooling for desktop PCs and

other types of computers.

7.16

7.17

US patent 4817709

Boundary layer regrowth

Flow transition or turbulence

FlowAir

Heat Transfer Enhancement

Interrupted surface

Performance Performance

x

average

Plain fin - continuous fin

average

x

Heat Transfer Enhancement

Heat Transfer Enhancement