Internal Flow 1

Chapter 8

Convection: Internal Flow (8.1-8.6)

Internal Flow 2

IntroductionIn Chapter 7 we obtained a non-dimensional form for the heat transfer coefficient, applicable for problems involving external flow: Calculation of fluid properties was done at surface temperature,

bulk temperature of the fluid, or film temperature

• In this chapter we will obtain convection coefficients for geometries involving internal flow, such as flow in tubes

Recall Newton’s law of cooling:

For flow inside a tube we cannot define Must know how temperature evolves inside the pipe and find alternative

expressions for calculating heat flux due to convection.

)(" TThq ss

T

Internal Flow 3

Flow Conditions for Internal Flow

• Onset of turbulent flow at 2300Re

DumD

• Hydrodynamic entry length: – Laminar flow

– Turbulent flow

Dhfd Dx Re05.0/,

10/, Dx hfd

0/ dxdu

Internal Flow 4

Mean Velocity

• Velocity inside a tube varies over the cross section. For every differential area dAc:

A

cA

dAxrumdm ),( cdAxrumd ),(

• Overall rate of mass transfer through a tube with cross section Ac:

cm Aum and c

m A

mu

where um is the mean (average) velocity

(8.1)

Combining with (8.1):

oc

r

oc

Ac

m rdrxrurA

dAxruu

02),(

2),(

Can determine average temperature at any axial location (along the x-direction), from knowledge of the velocity profile

(8.2)

Internal Flow 5

Velocity Profile in a pipe

• For laminar flow of an incompressible, constant property fluid in the fully developed region of a circular tube (pipe):

22 1

4

1)(

oo r

rr

dx

dpru

dx

dpru o

m

8

2

(8.3a)

(8.4)

2

12)(

om r

r

u

ru(8.3b)

Internal Flow 6

Thermal Considerations: Mean Temperature• We can write Newton’s law of cooling

inside a tube, by considering a mean temperature, instead of T

)("mSx TThq

• The internal energy per unit mass for a differential area is: ),()(),( xrTcudAxrTcmdEd c

• Overall rate of energy transfer :

cm

ETm

where Tm is the mean (average) velocity

Combining

with (8.6):

o

cr

om

Ac

m rdrxruTrucm

dAxrTucT

02),(

2),(

• Integrating over the entire cross section: cAA

dAxrTucEdE ),(

mTcmE and

(8.5)

(8.6)

(8.7)

Internal Flow 7

Example 8.1For flow of a liquid metal through a circular tube, the velocity and temperature profiles at a particular axial location may be approximated as being uniform and parabolic respectively. That is, u(r)=C1 and T(r)-Ts=C2[1-(r/ro)2], where C1 and C2 are constants and Ts the temperature at the surface of the tube. What is the value of the Nusselt number, NuD at this location?

])/(1[)( 22 os rrCTrT

1)( Cru ro

Ans. NuD=8

Internal Flow 8

Fully Developed Conditions

• For internal flows, the temperature, T(r), as well as the mean temperature, Tm always vary in the x-direction, ie.

0,0)(

dx

dT

dx

rdT m

?Can we claim that ? 0)(

dx

rdT

Internal Flow 9

Fully Developed Conditions

• A fully developed thermally region is possible, if one of two possible surface conditions exist :

– Uniform temperature (Ts=constant)

– Uniform heat flux (qx”=const)

• Thermal Entry Length :

10)/(

PrRe05.0)/(

,

,

turbtfd

Dlamtfd

Dx

Dx

• Although T(r) changes with x, the relative shape of the temperature profile remains the same: Flow is thermally fully developed.

0)()(

),()(

,

tfdms

s

xTxT

xrTxT

x

Internal Flow 10

Fully Developed Conditions

• It can be proven that for fully developed conditions, the local convection coefficient is a constant, independent of x:

)(xfh

Internal Flow 11

Mean temperature variation along a tubeWe are still left with the problem of knowing how the mean temperature Tm(x), varies as a function of distance, so that we can use it in Newton’s law of cooling to estimate convection heat transfer.

Consider an energy balance on a differential control volume inside the tube:

– Main contributions are due to internal energy changes [= ], convection heat transfer and flow work [=p], needed to move fluid.

mTcm

)( pTcdmdq mconv

The rate of convection heat transfer to the fluid is equal to the rate at which the fluid thermal energy increases, plus the net rate at which is work is done in moving the fluid through the control volume

P=surface perimeter

(8.8a)

Internal Flow 12

Mean temperature variation along a tubeConsidering perfect gas, or incompressible liquid:

)( ,, imompconv TTcmq

qconv is related to mean temperatures at inlet and outlet.

(8.9)

mpconv dTcmdq

By integrating:

(8.8b)

Combining equations 8.5 and 8.8b:

)("

mspp

sm TThcm

P

cm

Pq

dx

dT

where P=surface perimeter =

D for circular tube, width for flat plate

Integration of this equation will result in an expression for the variation of Tm as a function of x.

(8.10)

Internal Flow 13

Case 1: Constant Heat Flux

• Integrating equation (8.10):

)("" LPqAqq ssconv

xcm

PqTxT

p

simm

"

,)( (8.11)

where P=surface perimeter

D for circular tube,

=width for flat plate

constqs "

Internal Flow 14

Example (Problem 8.15)A flat-plate solar collector is used to heat atmospheric air flowing through a rectangular channel. The bottom surface of the channel is well insulated, while the top surface is subjected to a uniform heat flux, which is due to the net effect of solar radiation absorption and heat exchange between the absorber and cover plates.

For inlet conditions of mass flow rate=0.1 kg/s and Tm,i=40°C, what is the air outlet temperature, if L=3 m, w=1 m and the heat flux is 700 W/m2? The specific heat of air is cp=1008 J/kg.K

Ans: Tm,o=60.8°C

Internal Flow 15

Case 2: Constant Surface Temperature, Ts=constant

From eq.(8.10) with Ts-Tm=T: Thcm

P

dx

Td

dx

dT

p

m

)(

Integrating from x to any downstream location:

hcm

Px

TT

xTT

pims

ms

exp

)(

,

For the entire length of the tube:

hcm

PL

T

T

TT

TT

pi

o

ims

oms

exp

,

,

lmsconv TAhq where)/ln( io

iolm TT

TTT

(8.13) (8.14)

(8.12)

As is the tube surface area, As=P.L=DL

Internal Flow 16

Example 8.3

Steam condensing on the outer surface of a thin-walled circular tube of 50 mm diameter and 6-m length maintains a uniform surface temperature of 100°C. Water flows through the tube at a mass flow rate of 0.25 kg/s and its inlet and outlet temperatures are Tm,1=15°C and Tm,o=57°C. What is the average convection coefficient associated with the water flow?

D=50 mm

L=6 m

Ts=100°C

Tm,i=15°CTm,o=57°C

KmWhAns 2/756 .

Internal Flow 17

Case 3: Uniform External Temperature

Replace Ts by and by (the overall heat transfer coefficient, which includes contributions due to convection at the tube inner and outer surfaces, and due to conduction across the tube wall)

T h U

p

s

im

om

i

o

cm

AU

TT

TT

T

T

exp,

,

lms TAUq (8.15) (8.16)

Internal Flow 18

Summary (8.1-8.3)• We discussed fully developed flow conditions for cases involving

internal flows, and we defined mean velocities and temperatures• We wrote Newton’s law of cooling using the mean temperature,

instead of

• Based on an overall energy balance, we obtained an alternative expression to calculate convection heat transfer as a function of mean temperatures at inlet and outlet.

• We obtained relations to express the variation of Tm with length, for cases involving constant heat flux and constant wall temperature

T)("

mS TThq

)( ,, imompconv TTcmq

xcm

PqTxT

p

simm

"

,)(

hcm

PL

T

T

TT

TT

pi

o

ims

oms

exp

,

,

(8.9)

Internal Flow 19

Summary (8.1-8.3)

• We can combine equations (8.13-8.16) with (8.9) to obtain values of the heat transfer coefficient (see solution of Example 8.3)

In the rest of the chapter we will focus on obtaining values of the heat transfer coefficient h, needed to solve the above equations

• We used these definitions, to obtain appropriate versions of Newton’s law of cooling, for internal flows, for cases involving constant wall temperature and constant surrounding fluid temperature

lmsconv TAhq lms TAUq

)/ln( io

iolm TT

TTT

(8.13-8.16)