External Flow:External Flow:Flow over Bluff ObjectsFlow over Bluff Objects

(Cylinders, Spheres, Packed Beds) (Cylinders, Spheres, Packed Beds)andand

Impinging JetsImpinging Jets

Chapter 7Chapter 7

Sections 7.4 through 7.8Sections 7.4 through 7.8

Cylinder in Cross Flow Cylinder in Cross Flow

Circular Cylinder in Cross Flow• Conditions depend on special features of boundary layer development, including onset at a stagnation point and separation, as well as transition to turbulence.

– Stagnation point: Location of zero velocity and maximum pressure. 0u

– Followed by boundary layer development under a favorable pressure gradient and hence acceleration of the free stream flow . / 0dp dx / 0du dx

– As the fluid flows towards the rear of the cylinder, pressure increases. The boundary layer develops under the influence of an adverse pressure gradient:

/ 0, / 0 .dp dx du dx

Cylinder in Cross Flow (cont.)Cylinder in Cross Flow (cont.)

– Separation occurs when the velocity gradient becomes zero0/ ydu dy

followed by flow reversal and a wake downstream of the separation point.

Cylinder in Cross Flow (cont.)Cylinder in Cross Flow (cont.)

• Drag force on the cylinder is the sum of friction and form (pressure) drag

• Define drag coefficient:

Where FD = Total drag force Af = Frontal area of cylinder = Diameter x Length

2Figure 7.8

/ 2D

D

f

FC

A V

– Location of separation depends on boundary layer transition.

– For Reynolds number > 200,000 on a smooth cylinder, the boundary becomes turbulent. Flow separation is delayed, and the wake is small then that when the boundary is laminar.

ReD

VD VD

Cylinder in Cross Flow (cont.)Cylinder in Cross Flow (cont.)

Cylinder in Cross Flow: Experimental Drag Coefficient

Cylinder in Cross Flow (cont.)Cylinder in Cross Flow (cont.)

Cylinder in Cross Flow:Heat Convection Experiment

Cylinder in Cross Flow (cont.)Cylinder in Cross Flow (cont.)

Local Heat Convection Coefficient: Experimental Results

Turbulent FlowTurbulent Flow

Experimental Data for Average Nusselt Number / :DNu hD k

Cylinder in Cross Flow (cont.)Cylinder in Cross Flow (cont.)

– Correlation of Average Nusselt Number:1/ 3Re Prm

D DNu C

4 / 55 / 81/ 2 1/ 3

1/ 42 / 3

0.62Re Pr Re0.3 1

282,0001 0.4 / Pr

D DDNu

– Churchill and Bernstein correlation for all ReD :

Spheres and Packed BedsSpheres and Packed Beds

Forced Convection on Sphere

1/ 41/ 2 2 / 3 0.42 0.4Re 0.06Re Pr /D D D sNu

– Flow characteristics similar to flow over cylinder.

– Drag coefficient lower than that of cylinder because of 3D relief effect (Fig. 7.8)

– Correlation of average Nusselt number (s is the fluid viscosity at surface temperature):

Cylinder in Cross Flow (cont.)Cylinder in Cross Flow (cont.)

Cylinders of Noncircular Cross Section: 1/ 3Re PrmD DNu C

Tube BanksTube Banks

Flow Across Tube Banks• Typical design for two-fluid heat exchangers.

Tube BanksTube Banks

Definition of Max. Velocity

maxSTV V

S DT

Aligned:

Staggered:

if 2maxSTV V S D S DD TS DT

if 2max 2

STV V S D S DD TS DD

Tube Banks (cont.)Tube Banks (cont.)

• Aligned Tube Bank: Flow behind first row is turbulent, causing convection coefficients to increase gradually in subsequent rows if there are sufficiently spacings.

If the rows are too close (ST/SL < 0.7), each tube is confined in the wake of the one in front, and convection coefficient would decrease instead.

Tube Banks (cont.)Tube Banks (cont.)

• Staggered Tube Bank: Heat convection is more evenly distributed, and therefore more effective then the aligned tube bank.

Tube Banks (cont.)Tube Banks (cont.)

• Zukauskas Correlation for an Isothermal Array: 1/ 40.362 ,maxRe Pr Pr/ Prm

D D sNu C C

Tube Banks (cont.)Tube Banks (cont.)

Pr is evaluated at average of inlet and outlet temperature of the cross flow.

Prs is evaluated at surface temperature of the tubes Ts.

Reynolds number for the flow across the tube bank is defined as: ReD,max = Vmax D/

/ 2i oT T

Tube Banks (cont.)Tube Banks (cont.)

• From an energy balance, it can be shown that the inlet & outlet temperatures (Ti , To) of the flow is related by: (Challenge: Prove it!) • N is the total number of tubes:

s o

s i T T p

T T DNhexp

T T VN S c

T LN N N x

• Total Heat Convection Rate can also be shown as:

• As is the total surface area of the tubes:

• Tlm is called the “Log-Mean Temperature”:

s mq hA T

sA N DL

s i s om

s i

s o

T T T TT

T Tn

T T

• Correlation of Pressure Drop across the tube bank:

2max

2L

Vp N f

Tube Banks (cont.)Tube Banks (cont.)

Correlation Factor & Friction Factor: Aligned Tube Bank

Tube Banks (cont.)Tube Banks (cont.)

Correlation Factor & Friction Factor: Staggered Tube Bank

Spheres and Packed BedsSpheres and Packed Beds

Flow Through Packed Beds of Particles

– Large surface area per unit volume: desirable for the transfer and storage of thermal energy.

Spheres and Packed Beds (cont.)Spheres and Packed Beds (cont.)

– Correlation For a packed bed of spheres:

0.5752.06 Re

void fraction (0.3 < < 0.5)

H Dj

,p t mq hA T

, total surface area of particlesp tA

–

,

,

exp p ts o

s i c b p

hAT T

T T VA c

, cross-sectional area of bedc bA

–

– jH is the Colburn j-factor: jH = St Pr2/3

JetsJets

Jet Impingement• Characterized by large convection coefficients and used for cooling and heating in numerous manufacturing, electronic and aeronautic applications.

JetsJets

• Example of Multiple Jet Impingent Design: Slot Jet Array

JetsJets

– Relative Nozzle Area:

JetsJets

– Hydraulic diameter: Dh = 4 Area/Perimeter = D (round nozzle), Dh ~ 2W (slot nozzle)

– Typical Experimental Data of Local Nusselt number distribution:

– Average Nusselt number:

from Fig. 7.17

hr h

e hr

hDNu f Re, Pr, A ,H / D

kV D

Re , A

Correlations Section 7.7.2