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 ReRe

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 (μ∞/μs)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 Flow7.2.1 Laminar Flow Over an Isothermal Plate: A Similarity SolutionRead the textbook pp. 405-410.

With stream function ψ(x, y) and the transformation of η, f (η) 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 δ and the local friction coefficient Cf,x can be determined as (7.19) and (7.20).

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 thatThe average heat transfer coefficient is

Hence,

Similarly,

For small Pr, namely liquid metals, δt >>δ, we may assume u=u∞throughout the thermal boundary layer and obtain (7.32).For all Pr numbers: (7.33)

3/1

0332.0* Pr

ddT

==ηη

1/ 2 1/30.332 0.6xx x

h xNu Re Pr Prk

≡ = ≥

3/1Prt≈

δδ

xx

xx hdxhx

h 210 ==∫= K

1/ 2 1/30.664 0.6xx x

h xNu Re Pr Prk

≡ = ≥

, 1/ 2 1/3

AB

0.664 0.6m xx x

h xSh Re Sc Sc

D≡ = ≥

(7.23)

(7.24)

(7.30)

(7.31)

, 1/ 2, 2 0.664

/ 2s x

f x xC Reuτ

ρ−

∞

≡ = (7.20)

7.2.2 Turbulent Flow over an Isothermal PlateFrom 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 (δ 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 −= xxReδ

δ ≈ δ t ≈ δc

4/5 1/30.0296 0.6< <60x x xNu St Re Pr Re Pr Pr= =4/5 1/30.0296 0.6< <3000x m x xSh St Re Sc Re Sc Sc= =

(7.34)

(7.35)

(7.36)

(7.37)

7.2.3 Mixed boundary Layer ConditionsWhen 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 LL

AC Re Re ReRe

− ⎡ ⎤= − ≤ ≤⎣ ⎦

4/5 1/38

0.6 60(0.037 )

10L L

x,c L

PrNu Re A Pr

Re Re≤ ≤⎡ ⎤

= − ⎢ ⎥≤ ≤⎣ ⎦

4/5 1/38

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. 7.4):

Laminar flow:

turbulent flow:

01/33/ 41 ( / )

xx

NuNu

xξ

ξ==

⎡ ⎤−⎣ ⎦

01/99/101 ( / )

xx

NuNu

xξ

ξ==

⎡ ⎤−⎣ ⎦

(7.42)

(7.43)

7.2.5 Flat Plates with Constant Heat Flux ConditionsFor 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 .

"

( ) ss

x

qT x Th∞= +

"

0 0

"

1/ 2 1/3

1 ( ) ( )

where 0.680

L Lss s

x

s

L

L L

q xT T T T dx dxL L kNu

q Lk Nu

Nu Re Pr

∞ ∞− = − =

=

=

∫ ∫

1/ 2 1/30.453 , 0.6xx x

h xNu 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 CoefficientsErrors 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 CalculationFollow 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 FlowBoundary 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 (≣ρVD/μ).

Drag coeff. CD→Fig. 7.8

7.4.2 Convection Heat and Mass TransferThe local Nuθ → Fig. 7.9

Rise due to mixing in the wake

Decline due boundary layer growth

Rise due to transition to turbulence

The average (of more engineering interest):

Constants for circular cylinders: Table 7.2

1/3mD D

hDNu CRe Prk

≡ = --empirical (7.52)

7.52

Constants for noncircular cylinders: Table 7.3

1/3mD D

hDNu CRe Prk

≡ = (7.52)

Other correlations are shown in (7.53), (7.54).EX 7.4

52

7.5 The SphereWhitaker:

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

EX 7.6

1/41/ 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

PrRe

μμ

μ μ

⎛ ⎞= + + ⎜ ⎟

⎝ ⎠≤ ≤⎡ ⎤

⎢ ⎥≤ ≤ ×⎢ ⎥⎢ ⎥≤ ≤⎣ ⎦

(7.56)

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

7.6 Flow across Banks of Tubes (brief introduction)

Aligned or staggered: Fig. 7.11 and Fig. 7.12A 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 smallReD (<100).Since the fluid temperature may change over the tube bank, the heat transfer rate may be significantly over-predicted by using ΔT=Ts-T∞. The appropriate form of ΔT is a log-mean temperature difference ΔTlm, as shown in (7.66). And

(7.68)

The pressure drop (7.69))(' lmTDhNq Δ= π

DNu

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.

US patent 4817709

λ

Boundary layer regrowthFlow 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