Scaling of Effectiveness at a Design Point to Off Design Conditions

Author: Peter Martinello

Supervisory Committee:

Dr. William Lear

Dr. Sanim Anghaie

Dr. S.A. Sherif University of FloridaDepartment of Mechanical and Aerospace EngineeringNovember 28, 2005

Introduction

• Purpose– To obtain a model that will predict off design

effectiveness of a given exchanger

• Motivation– VARS integration– Modeling for shell recuperator

• Scope– Determine dimensionless parameters that affect ε/εD

– Determine how those parameters affect ε/εD

– Determine how Re affects ε/εD

Background Research

• Read Papers on How Re Affects Effectiveness and Heat Transfer– Geometry Based Studies– Approach

• Reviewed General Heat Exchanger Theory– NTU Method– Temperature Curves & HX Control Volumes

• Internal Flow– Regions and Regimes– Convection Correlation Equations

Analysis

• Initial Assumptions– Shell Counter Flow– Full Developed (Thermal Entrance Region Ignored)– Thin-walled (Conduction Term Ignored)

• Starting Points– ε = q / qmax = m Cp (Ti – To) /[(m Cp)min (Th,i-Tc,i)]– q = P ∫ q’’(x) dx– P q’’ dx = Cp dT –

Analysis (cont.)

• Intermediate Steps

–

–

–

Analysis (cont.)

• Final Results of Analysis

–

–

Results

• Plotting ε/εD as a function of r– NTU & NTUD held const.

– rD varied

• Qualitative Meaning– Min ε/εD occurs when r=1

– As rD increases ε/εD increases

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Effectiveness Ratio vs r with NTU and NTUD

both = 2.5

e /

eD

r

rD

= 1 & effd = 0.71

rD

= 0.75 & effd = 0.78

rD

= 0.5 & effd = 0.83

rD

= 0.25 & effd =0.88

rD

= 0 & effd =0.92

Results (cont.)

• Relationship of ε/εD as NTU is increased or decreased as r varies– As NTU increases minimizing effect on ε/εD as r increases is

retarded and than more sharply realized– For large values of NTU it dominates the exponential longer

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Effectiveness Ratio vs r with NTU and NTUD

both = 1

e /

eD

r

rD

= 1 & effd = 0.50

rD

= 0.75 & effd = 0.53

rD

= 0.5 & effd = 0.56

rD

= 0.25 & effd =0.60

rD

= 0 & effd =0.63

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Effectiveness Ratio vs r with NTU and NTUD

both = 4

e /

eD

r

rD

= 1 & effd = 0.80

rD

= 0.75 & effd = 0.87

rD

= 0.5 & effd = 0.93

rD

= 0.25 & effd =0.96

rD

= 0 & effd =0.98

Results (cont.)

• Initial Obstacle– Convection

coefficient in transition regime

• Solution– Use weighted

averages in the transition regime

– Purple path is the path used for convection coefficient

0 2000 4000 6000 8000 100000

1

2

3

4

5

6

7

8

9Convection Coefficient vs Re

h

Re

laminar - lamturbulentweighted average - waaverage lam & wah as a f(Re)

Results (cont.)• Plotting ε/εD as a function of Re

– Getting NTU and r in terms of Re

• NTU = UA/(mCp)min – Convection Correlations in terms of Re

• r = (mCp)min / (mCp)max

– Mass flow in terms of Re

• Assumptions– Air-to-air shell HX, rD and NTUD are const., Di/Do=.5,

Temp const. from design to arbitrary, only Rei varies

i = ¼ π Di μ Rei o = ¼ π (Do + Di) μ Reo

Results (cont.)

• Plot ε/εD, r, NTU in terms of Re– Starts at a maximum

at very low values of Re

– Then decreases as Re increase, r increases

– Reached minimum at Re where r=1, agrees with other plot

– Starts to increase again past peak of r

– Approaches new local max, Cmin is much larger at this point

101

102

103

104

105

106

0

5

8

10

15

Effectiveness Ratio, NTU and r vs Re effd = 0.08

eff/

eff d

, 8r,

(1

/6)N

TU

Re

eff/effd

8r(1/6)NTU

Results (cont.)

• NTU in terms of Re– NTU follows

same curve as ε/εD except for very low Re

– At very low Re Cmin is extremely small

101

102

103

104

105

106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

NTU and r vs Re effd = 0.08

r &

(1

/6)N

TU

Re

(1/6)NTUr

Conclusion

• ε/εD

– Decreases as r increases– Increases as rD increases

• All plotted results agree with this

– At larger values of NTU the affect of increasing r is retarded

– Is minimum at Re where r=1– Hits a local max at very large values of Re

Conclusion (cont.)

• Recommendations– Examine scenarios where both Re varies– Develop a better model for overall heat transfer

coefficient, convection coefficients, entry region– Take experimental data and match to results obtained

with mathematical models

• Applying This Work– Many known equations for effectiveness for various HX– Apply those equations to procedure and code to

extend this work beyond counter flow shell HX