Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

1

Forced Convection in Internal Geometries

No free flow condition (u∞ and T∞) exists in this case:

Flow is determined by two boundary layers

at entrance: homogeneous velocity profile

after hydrodynamic entrance length: fully developed velocity profile (no changes with x anymore)

Chap. 14: Internal Convection

x xd,h

r0

fd, x

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

2

Pipe Flow – Hagen-Poiseuille Equation

Chap. 14.1: Pipe Flow

( ) ( ) ( )220

21220 4

14

)( rrdxdp

lpprrru −⋅

⋅⋅−=

⋅⋅

−−=

µµ

This leads to the following velocity profile u(r):

µ⋅⋅−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−⋅=

41)(

20

max20

2

maxr

dxdpuwith

rruru

The Hagen–Poiseuille equation describes a fluid flowing through a long cylindrical pipe due to a pressure drop. The assumptions of the equation are that the flow is laminar and incompressible and the flow is through a constant circular cross-section that is substantially longer than its diameter.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

3

dAruA

uA

m ⋅∫= )(1

Definition of a mean velocity in the pipe:

AuV m ⋅=

Using the result for u(r):

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⋅=→=⋅

⋅−=

2

0

max20 12)(

28 rruruu

dxdpru mm µ

Mean velocity:

Chap. 14.1: Pipe Flow

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

4

∫ ⋅⋅=⋅⋅=A

m dArxuAum ),(ρρ

Mass flow rate the entrance region: u = u(x,r)

( )∫ ⋅⋅⋅=

⋅⋅

∫ ⋅⋅⋅⋅=

⋅=

0

0

020

20

0 ),(2,2 r

r

m drrrxurr

drrrxu

Amu

πρ

ρπ

ρ

Using the pipe geometry:

Using mass conservation (incompressible fluid) : um ≠ f(x)

2300ReRe , ≈⋅⋅

= kritDm

DDu

µρTurbulence for

ReD > ReD,krit

Chap. 14.1: Pipe Flow

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

5

Dlam

hfd

DX

Re05.0, ⋅≈⎟⎟⎠

⎞⎜⎜⎝

⎛

6010 , ≤⎟⎟⎠

⎞⎜⎜⎝

⎛≤

turb

hfd

DX

Hydrodynamic entrance length (semi-empirical):

Laminar flow:

Turbulent flow:

flow)laminar (forRe64

2/)(

2 22

Dmm u

DdxdpfufDdxdp

=⋅

⋅−≡⋅⋅=⋅−

ρρ

Moody Friction Factor:

For laminar flow: Use results from Hagen-Poiseuille equation:

Chap. 14.1: Pipe Flow

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

6

Moody Diagram

Chap. 14.1: Pipe Flow

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

7

Thermal Considerations

Considering pipe flow with heat transfer between fluid and wall:

PrRe05.0, ⋅⋅≈⎟⎟⎠

⎞⎜⎜⎝

⎛D

lam

tfd

DX

Thermal entrance length (laminar):

Chap. 14.2: Pipe Flow – Thermal Considerations

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

8

)(),()( xTcmdArxTcruE mpA

pconv ⋅⋅≡⋅⋅⋅⋅= ∫ ρ

( )p

Ap

m cm

dArxTcruxT

⋅

⋅⋅⋅⋅

=∫

),()(ρ

drrrxTruru

xTr

mm ∫ ⋅⋅⋅

⋅=

0

020

),()(2)(

( )ms TThq −⋅=ʹ′ʹ′

Definition of mean temperature Tm using the flow enthalpy:

solved for Tm:

Mass flow rate expressed by um :

Convective heat transfer depending on Tm:

Note: For pipe flow, um und Tm are used instead of u∞ and T∞

Chap. 14.2: Pipe Flow – Thermal Considerations

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

9

)()(..0),(),( xfreiTTxrTT

xxxr

ms

s ≠==⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−

∂

∂=

∂

∂θθ

θ

ms

s

TTxrTT

xr−

−=

),(),(θ

!0)( ≠=dxdTandxTT m

mm

Is the temperature profile stationary / fully developed ?

The absolute temperature changes along x:

Introduction of a dimensionless temperature:

The ‘relative shape‘ of the temperature profile is constant:

Chap. 14.2: Pipe Flow – Thermal Considerations

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

10

Energy Balance for volume element

Goal: Temperature profile of Tm(x) for known boundary conditions and h value

Assumptions for derivation: - Pipe flow, steady-state: - ΔKE = ΔPE = 0 - no heat conduction in axial direction

.constm=

Chap. 14.3: Pipe Flow – Energy Balance

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

11

( ) ( )

( ) mpmvs

mvmvmvs

dTcmvpTcdmdq

dxdx

vpTcdmvpTcmvpTcmdq

⋅⋅=⋅+⋅⋅=→

=⎥⎦

⎤⎢⎣

⎡ ⋅+⋅+⋅+⋅⋅−⋅+⋅⋅+

0)(

Entering heat

Entering enthalpy flux

Leaving enthalpy flux + - = 0

First Law for open system for volume element (δW = 0):

After integration over entire pipe length: First Law for pipe

)( ,, inmoutmps TTcmq −⋅⋅=

Valid for compressible and incompressible fluids

Heat transfer through pipe wall:

Chap. 14.3: Pipe Flow – Energy Balance

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

12

( )mspp

m

mps

TThcmPq

cmP

dxdT

dTcmdxPqdq

−⋅⋅⋅

=ʹ′ʹ′⋅⋅

=→

⋅⋅=⋅⋅ʹ′ʹ′=

Using heat flux through the outer surface of the control volume (P = inner pipe perimeter)

ODE for the development of Tm along x - direction

2 specific cases are of interest:

•  Heat flux is constant

•  Inner wall temperature is constant

Chap. 14.3: Pipe Flow – Energy Balance

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

13

sp

m qcmP

dxdT

ʹ′ʹ′⋅⋅

=

xcmPqTxTp

sinmm ⋅

⋅

⋅ʹ′ʹ′+=,)(

Solution for constant heat flux through pipe wall

ODE to be solved:

Boundary condition: Tm(0) = Tm,in

Tm increases linearly

Entrance length: h high, (Ts - Tm) small

No quantitative result for h possible yet !

Chap. 14.3.1: Pipe Flow – Constant Heat Flux

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

14

)( msp

m TThcmP

dxdT

−⋅⋅⋅

=

ThcmP

dxTd

p

Δ⋅⋅⋅

=Δ

−

)(

( )⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅

⋅−=

Δ

Δ∫∫

Δ

Δ

L

p

T

T

dxhLcm

LPTTdout

in 0

1

Solution for constant wall temperature

ODE to be solved: formal analogy to transient conduction (0-D)

Homogeneous ODE by using new variable ΔT = Ts - Tm

Separation of variables, integration over entire pipe length

Chap. 14.3.2: Pipe Flow – Constant Wall Temp.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

15

∫ ⋅=L

L dxhL

h0

1L

pin

out hcmLP

TT

⋅⋅

⋅−=

Δ

Δ

ln

Using average h value:

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅

⋅−=

−

−=

Δ

ΔL

pinms

outms

in

out hcmLP

TTTT

TT

exp

,

,

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅

⋅−=

−

−x

pinms

ms hcmxP

TTxTT

exp)(

,

hhh Lx ==

Rearranging and using absolute temperatures:

Mean fluid temperature at any location x:

After thermal entrance length, h is constant. Therefore, for short entrance lengths:

Chap. 14.3.2: Pipe Flow – Constant Wall Temp.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

16

ThLPqs Δ⋅⋅⋅=

Total heat transfer calculated with averaged values:

p

xin

L

cmPhandeTxTwhere

dxxTL

T

⋅

⋅=⋅Δ=Δ

⋅Δ⋅=Δ

⋅−

∫

γγ)(:

)(1

0

Average temperature difference:

Chap. 14.3.2: Pipe Flow – Constant Wall Temp.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

17

( )10

−⋅⋅

Δ−=⋅

⋅

Δ−=Δ ⋅−⋅− LinLxin e

LTe

LTT γγ

γγ

⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ

Δ=⋅→Δ=⋅Δ ⋅−

out

inout

Lin T

TLTeT lnγγ

⎟⎠⎞

⎜⎝⎛

ΔΔ

Δ−Δ=Δ

in

out

inout

TT

TTTln

Using function for ΔT(x):

Substituting with ΔT at outlet → rearranging

Log Mean Temperature Difference: (for known inlet and outlet temperatures)

Chap. 14.3.2: Pipe Flow – Constant Wall Temp.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

18

⎟⎠⎞

⎜⎝⎛

ΔΔ

Δ−Δ⋅⋅=Δ⋅⋅=

in

out

inoutLLs

TT

TTLUTLUqln

If instead of the wall temperature the outer surrounding fluid temperature is known:

The convective heat transfer is substituted by a total heat transfer coefficient (h → UL): linear tube U-value !

Chap. 14.3.2: Pipe Flow – Constant Wall Temp.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

19

Heat transfer coefficients for laminar pipe flows

We need to solve for the temperature gradient on the wall surface:

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅=⋅+⋅

rTr

rrrTv

xTu

∂∂

∂∂α

∂∂

∂∂

If u and v are known: ODE can be solved for laminar pipe flow in the fully developed region:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⋅=

2

0

12)(rruru m00 =

∂

∂=

xuandv

Chap. 14.3.3: Pipe Flow – Heat Transfer Coeff.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

20

Heat conduction in axial direction is negligible compared to convection: assumption already used for Tm(x)

02

2=

∂

∂

xT

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟

⎠

⎞⎜⎝

⎛⋅=⎟

⎠

⎞⎜⎝

⎛ ⋅⋅2

0

121rr

dxdTu

drdTr

drd

rmm

α

Using solution for u(r):

For case: known = constant , the temperature gradient is known = constant

sq ʹ′ʹ′

dxdTm

Conduction radial

Convection axial

Chap. 14.3.3: Pipe Flow – Heat Transfer Coeff.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

21

2120

42

120

42

)ln(164

2)(

422

CrCrrr

dxdTurT

Crrr

dxdTu

drdTr

mm

mm

+⋅+⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅−⋅⎟

⎠

⎞⎜⎝

⎛⋅=

+⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅−⋅⎟

⎠

⎞⎜⎝

⎛⋅=⋅

α

α

Integrating twice:

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⎟⎠

⎞⎜⎝

⎛⋅−=→=

1632)(),(

20

20r

dxdTuTCxTxrT mm

ss α

Boundary conditions:

T(0) is finite / symmetry at x = 0 → C1 = 0

T(r0) = Ts(x) is given (directly or through heat flux)

Chap. 14.3.3: Pipe Flow – Heat Transfer Coeff.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

22

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+⋅⎟

⎠

⎞⎜⎝

⎛⋅⋅⋅

−=2

0

4

0

20

41

161

1632)(

rr

rr

dxdTruTrT mm

s α

Entire temperature profile:

drrrTruru

Tr

mm ∫ ⋅⋅⋅

⋅=

0

020

)()(2( )xTcuE mpmconv ⋅⋅⋅≡ ρ

Definition of mean temperature Tm (see above):

u(r) and T(r) are known, resulting in:

dxdTruTT mm

sm ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⋅−=

α

20

4811

Chap. 14.3.3: Pipe Flow – Heat Transfer Coeff.

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

23

Temperature profile: Gradient is almost constant over half of cross-section

Additional analysis for constant heat flux through pipe wall

00

20

22ruc

qdxdTrq

dxdTruc

mp

sms

mmp ⋅⋅⋅

ʹ′ʹ′⋅=→⋅⋅ʹ′ʹ′=⋅⋅⋅⋅⋅ρ

ππρ

Axial temperature increase driven by heat flux:

Chap. 14.3.3: Pipe Flow – Heat Transfer Coeff.

ϕTs

Tm

D

Tem

pera

tur

Rohrdurchmesser-r0 +r0 0

DTT ms −⋅=

1148)tan(ϕ

Pipe radius

Tem

pera

ture

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

24

Using this result, we can calculate h:

36.41148

=⋅

=→=kDhNu

Dkh D

Valid for laminar pipe flow and = constant sq ʹ′ʹ′

Constant wall temperature (derivation not shown)

.66.3 constTNu sD ==

Chap. 14.3.3: Pipe Flow – Heat Transfer Coeff.

hq

kDqTT ss

smʹ′ʹ′

=⋅ʹ′ʹ′

⋅−=−4811Difference between wall and

mean fluid temperature: f(h)

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

25

Summary for Laminar Pipe Flow

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⋅=

2

0

12)(rruru mVelocity profile

ThLPqs Δ⋅⋅⋅=Heat transfer through pipe wall

Heat flux constant: Wall temperature constant:

qLPqs ʹ′ʹ′⋅⋅=

hqTʹ′ʹ′

=Δ

66.3=DNu36.4

1148

=→= DNuDkh

⎟⎠⎞

⎜⎝⎛

ΔΔ

Δ−Δ=Δ

in

out

inout

TT

TTTln

p

Linout cm

PhwitheTT⋅

⋅=⋅Δ=Δ ⋅−

γγ

Chap. 14.4: Pipe Flow – Summary

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

26