Internal Flow:General Considerations

Chapter 8Sections 8.1 through 8.3

Entrance Conditions

Entrance Conditions• Must distinguish between entrance and fully developed regions.

• Hydrodynamic Effects: Assume laminar flow with uniform velocity profile at inlet of a circular tube.

– Velocity boundary layer develops on surface of tube and thickens with increasing x.

– Inviscid region of uniform velocity shrinks as boundary layer grows. Does the centerline velocity change with increasing x? If so, how does it change?

– Subsequent to boundary layer merger at the centerline, the velocity profile becomes parabolic and invariant with x. The flow is then said to be

hydrodynamically fully developed. How would the fully developed velocity profile differ for turbulent flow?

Entrance Conditions (cont)

• Thermal Effects: Assume laminar flow with uniform temperature, , at inlet of circular tube with uniform surface temperature, , or heat flux, .

,0 iT r Ts iT T sq

– Thermal boundary layer develops on surface of tube and thickens with increasing x.

– Isothermal core shrinks as boundary layer grows.

– Subsequent to boundary layer merger, dimensionless forms of the temperature profile become independent of x. for and s sT q Conditions are then said to be

thermally fully developed.

Is the temperature profile invariant with x in the fully developed region?

Entrance Conditions (cont)

For uniform surface temperature, what may be said about the change in the temperature profile with increasing x?

For uniform surface heat flux, what may be said about the change in the temperature profile with increasing x?

How do temperature profiles differ for laminar and turbulent flow?

Mean Quantities

The Mean Velocity and Temperature• Absence of well-defined free stream conditions, as in external flow, and hence a reference velocity or temperature , dictates the use of a cross- sectional mean velocity and temperature for internal flow.

u T

mu mT

or,

,cA cm u r x dA

Hence,

,cA c

mc

u r x d Au

A

orFor incompressible flow in a circular tube of radius ,

2

2 ,or

m oo

u u r x r drr

• Linkage of mean velocity to mass flow rate:

m cm u A

Mean Quantities (cont)

• Linkage of mean temperature to thermal energy transport associated with flow through a cross section:

ct A c mE uc T dA mc T

Hence,

cA cm

uc T dAT

mc

• For incompressible, constant-property flow in a circular tube,

02

2 , ,or

mm o

T u x r T x r r dru r

• Newton’s Law of Cooling for the Local Heat Flux:

s s mq h T T

What is the essential difference between use of for internal flow and for external flow?

mT T

Entry Lengths

Hydrodynamic and Thermal Entry Lengths

• Entry lengths depend on whether the flow is laminar or turbulent, which, in turn, depend on Reynolds number.

Re m hD

u D

The hydraulic diameter is defined as

4 ch

ADP

in which case,

4Re m hD

u D mP

For a circular tube,

4Re mD

u D mD

Entry Lengths (cont)

– Onset of turbulence occurs at a critical Reynolds number of

,Re 2300D c

– Fully turbulent conditions exist for

Re 10,000D

• Hydrodynamic Entry Length

,Laminar Flow: / 0.05Refd h Dx D

,Turbulent Flow: 10 / 60fd hx D

• Thermal Entry Length ,Laminar Flow: / 0.05 Re Prfd t Dx D

,Turbulent Flow: 10 / 60fd tx D

• For laminar flow, how do hydrodynamic and thermal entry lengths compare for a gas?An oil? A liquid metal?

Fully Developed Flow

Fully Developed Conditions• Assuming steady flow and constant properties, hydrodynamic conditions, including the velocity profile, are invariant in the fully developed region.

What may be said about the variation of the mean velocity with distance from the tube entrance for steady, constant property flow?

• The pressure drop may be determined from knowledge of the friction factor f, where,

2

// 2m

dp dx Df

u

Laminar flow in a circular tube:64

ReD

f

Turbulent flow in a smooth circular tube:

20.790 1n Re 1.64Df

Fully Developed Flow (cont)

Turbulent flow in a roughened circular tube:

Pressure drop for fully developed flow from x1 to x2:

2

1 2 2 12mu

p p p f x xD

and power requirementp mP p

Fully Developed Flow (cont)

• Requirement for fully developed thermal conditions:

,

,0s

s m fd t

T x T r xx T x T x

• Effect on the local convection coefficient:

/

o

o

r rs

s m s mr r

T rT Tf x

r T T T T

Hence, assuming constant properties,

/s

s m

q k h f xT T k

h f x

Variation of h in entrance and fully developed regions:

Mean Temperature

Determination of the Mean Temperature• Determination of is an essential feature of an internal flow analysis. mT x

Determination begins with an energy balance for a differential control volume.

conv m p mdq md c T p mc dT

Why is the second equality in the foregoing expression considered to be approximate?

Integrating from the tube inlet to outlet,

, , (1)conv p m o m iq mc T T

Mean Temperature (cont)

A differential equation from which may be determined is obtained bysubstituting for

mT x .

conv s s mdq q P dx h T T P dx

2m ss m

p p

dT q P P h T Tdx mc mc

• Special Case: Uniform Surface Heat Flux

m s

p

dT q Pf x

dx mc

,s

m m i

p

q PT x T x

mc

Why does the surface temperature vary with x as shown in the figure?In principle, what value does Ts assume at x=0?

Total heat rate:

conv sq q PL

Mean Temperature (cont)

• Special Case: Uniform Surface Temperature

From Eq. (2), with s m

m

p

T T T

d Td T P h Tdx dx mc

Integrating from x=0 to any downstream location,

,

exps mx

s m ip

T T x Px hT T mc

1x x

xoh h dx

x

Overall Conditions:

,

,

exp exps m oo s

i s m ip p

T TT h APLh

T T T mc mc

conv s mq h A T

3

1n /o i

mo i

T TT

T T

Mean Temperature (cont)

• Special Case: Uniform External Fluid Temperature

,

,

1exp expm oo s

i m ip p tot

T TT U AT T T mc mc R

ms m

tot

Tq UA TR

Eq. (3) with replaced by .m sT T T

Note: Replacement of by Ts,o if outer surface temperature is uniform.T

Problem: Water Flow Through Pipe in Furnace

Problem 8.17: Estimate temperature of water emerging from a thin-walled tube heated by walls and air of a furnace. Inner and outer

convection coefficients are known.

D = 0 .25 m , L = 8 m , = 1 Tm ,o T = 300 Km ,i

Water

AirT = 700 K oo

T fu r = 700 Kqcv,o q radT t

.m = 5 kg/s

T oo T oo

T t

R cv,i

R radR cv,o

T fu r =h = 50 W /m -Ko

2

KNOWN: Water at prescribed temperature and flow rate enters a 0.25 m diameter, black thin-walled tube of 8-m length, which passes through a large furnace whose walls and air are at a temperature of Tfur = T = 700 K. The convection coefficients for the internal water flow and external furnace air are 300 W/m2K and 50 W/m2K, respectively.

FIND: The outlet temperature of the water, Tm,o.

Problem: Water Flow Through Pipe in Furnace (cont)

D = 0 .25 m , L = 8 m , = 1 Tm ,o T = 300 Km ,i

Water

AirT = 700 K oo

T fu r = 700 Kqcv,o q radT t

.m = 5 kg/s

T oo T oo

T t

R cv,i

R radR cv,o

T fu r =h = 50 W /m -Ko

2

SCHEMATIC:

ASSUMPTIONS: (1) Steady-state conditions; (2) Tube is small object with large, isothermal surroundings; (3) Furnace air and walls are at the same temperature; and (3) Tube is thin-walled with black surface.

PROPERTIES: Table A-6, Water: cp ≈ 4180 J/kgK.

ANALYSIS: The linearized radiation coefficient may be estimated from Eq. 1.9 with = 1,

2 2rad t fur t furh T T T T

where tT represents the average tube wall surface temperature, which can be estimated from an energy balance on the tube.

As represented by the thermal circuit, the energy balance may be expressed as m t t fur

cv,i cv,o rad

T T T TR 1/ R 1/ R

The thermal resistances, with As = PL = DL, are

cv,i i s cv,o o s rad radR 1/ h A R 1/ h A R 1/ h

Problem: Water Flow Through Pipe in Furnace (cont)

and the mean temperature of the water is approximated as m m,i m,oT T T / 2

The outlet temperature can be calculated from Eq. 8.46b, with Tfur = T,

T T 1expT T

m c R

m,om,i

p tot

where

tot cv,icv,o rad

1R R1/ R 1/ R

with 5 4 4

cv,i cv,o radR 6.631 10 K / W R 3.978 10 K / W R 4.724 10 K / W

it follows that

m tT 331 K T 418 K m,oT 362 K <