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 Florida Department of Mechanical and Aerospace Engineering November 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/ε D –Determine how those parameters affect ε/ε D –Determine how Re affects ε/ε D

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/ε D as a function of r –NTU & NTU D held const. –r D varied Qualitative Meaning –Min ε/ε D occurs when r=1 –As r D increases ε/ε D increases

Results (cont.) Relationship of ε/ε D as NTU is increased or decreased as r varies –As NTU increases minimizing effect on ε/ε D as r increases is retarded and than more sharply realized –For large values of NTU it dominates the exponential longer

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

Results (cont.) Plotting ε/ε D 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, r D and NTU D are const., D i /D o =.5, Temp const. from design to arbitrary, only Re i varies i = ¼ π D i μ Re i o = ¼ π (D o + D i ) μ Re o

Results (cont.) Plot ε/ε D, 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, C min is much larger at this point

Results (cont.) NTU in terms of Re –NTU follows same curve as ε/ε D except for very low Re –At very low Re C min is extremely small

Conclusion ε/ε D –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