1

3. Internal Flow General Concepts:

2300Re4Re , ==== crDmm

D DmDuDuμπνμ

ρ &

⎪⎪⎩

⎪⎪⎨

⎧

>>

<<<

turbulentfully000,10Returbulent4000Re

altransition4000Re2300laminar2300Re

:RegimesFlow

D

D

D

D

(d)

2

(e)

Figure 1 Boundary layer development for laminar flow in a circular tube: (a) The hydrodynamic boundary layer and velocity profiles. (b) The thermal boundary layer and temperature profiles for surface thermal condition: constant temperature, Ts. (The fluid in the tube is being warmed.) (c) Velocity and temperature profiles for determining the mean (average) temperature at a location x. (d) The development of the thermal boundary layer in a tube. (The fluid in the tube is being cooled.) (e) Velocity profile in turbulent flow. - Hydrodynamic and thermal entry lengths: For laminar flow:

lamhfdDlamtfd

Dlamhfd

xDx

Dx

)(PrPrRe05.0)(

Re05.0)(

,,

,

=≈

≈

For turbulent flow:

Dxx

DxD

turbhfdturbtfd

turbhfd

10)()(

60)(10

,,

,

=≈

≤≤

Figure 2Variation of the friction factor and the convection heat transfer coefficient in the flow direction for flow in a tube (Pr>1)

3

Figure 3 Variation of local Nusselt number along a tube in turbulent flow for both uniform surface

temperature and uniform surface heat flux [Deissler (1953)]. Note: Nusselt number is insensitive to the type of thermal boundary condition in turbulent flow, and the turbulent flow correlations can be used for either type of boundary condition. The Mean Temperature:

Figure 4 Actual and idealized temperature profiles for flow in a tube (the rate at which energy is

transported with the fluid is the same for both cases). Note: Unlike the mean velocity, the mean temperature Tm will change in the flow direction whenever the fluid is heated or cooled. - Conservation of energy principle:

cmp

cp

p

pm

cppmp

AVCdAVTC

CmmTC

T

dAVTCmTCTCm

ρ

ρδ

ρδ

∫∫∫∫

==

==

&

&

&& )(

Assume: 1. Constant density and specific heat 2. A circular pipe of radius R: drrdARA cc ππ 2and2 ==

drrrxVrxTVR

Tm

m ∫= ),(),(22

- The bulk mean fluid temperature:

2,, emim

bTT

T+

=

4

Figure 5 Thermally fully developed flow characteristics for constant surface temperature

Heating: Relative shape of the temperature profile remains unchanged in the flow direction (x2>x1).

Hydrodynamically fully developed:

)(0),( rVVx

xrV=→=

∂∂

Thermally fully developed:

0)()(),()(

=⎥⎦

⎤⎢⎣

⎡−−

∂∂

xTxTxrTxT

x ms

s

Thermal Analysis:

Figure 6 The heat transfer to a fluid flowing in a tube.

Overall Tube Energy Balance:

)( ,, imemp TTCmQ −= && Newton’s Law of Cooling:

)( msxs TThq −=&

Figure 7 Energy interactions for a differential control volume in a tube.

Energy Balance on a Differential Control Volume:

mp dTCmQ && =δ

5

)( dxpqdAqQ sps &&& ==δ

)( msxpp

sm TThCmP

CmPq

dxdT

−==&&

&

1. Constant Surface Heat Flux ( constant=sq& ):

constant==p

sm

CmPq

dxdT

&

&

By integration:

xCmPq

TxTp

simm &

&+= ,)(

x

sms h

qxTxTxT

&=−=Δ )()()(

Figure 8 Variation of the tube surface and the mean fluid temperatures along the tube for the case of

constant surface heat flux. 1. Constant Surface Temperature ( constant=sT ):

)()( xTTxT ms −=Δ

dxhCmP

xTxTdxTh

CmP

xdxTd

dxxdT

xp

xp

m

&&−=

ΔΔ

→Δ=Δ

−=)()()()()(

By integration:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−−

→−=ΔΔ

p

Px

ims

msx

pi

x

CmAh

TTxTT

hCmxP

TT

&&exp

)(ln

,

Number of Transfer Unit (NTU):

p

P

CmAhNTU

&=

Note: For sem TTNTU ≈→> ,5

6

Figure 9 The variation of the mean fluid temperature along the tube for the case of constant temperature.

lmP TAhQ Δ=& Log Mean Temperature Difference (LMTD):

i

o

iolm

TT

TTT

ΔΔΔ−Δ

=Δln

Convection Correlations for Tubes: Fully Developed Region Table 1 Summary of Forced Convection Heat Transfer Correlations for Internal Flow in Smooth Circular

Tubes

Note: In many applications the tube length will exceed the thermal entry length, it is often reasonable to assume that the average Nusselt number for the entire tube is equal to the value associated with the fully developed region: fdDD NuuN ,≈ Flow in Noncircular Tubes Hydraulic Diameter:

PA

D ch

4=

Where P is wetted perimeter.

7

Note: 1. Hydraulic diameter that should be used in calculating the Reynolds and Nusselt number. 2. The approach can be used for both laminar and turbulent flows.

Table 2 Nusselt Numbers for Fully Developed Laminar Flow in Noncircular Tubes

8

Example 1:

9

Example 2:

10

11

Example 3:

12

13

Example 4:

14