Convection Heat Transfer Experiment

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1 Convection Heat Transfer Experiment Thermal Network Solution with TNSolver Bob Cochran Applied Computational Heat Transfer Seattle, WA ME 331 Introduction to Heat Transfer University of Washington October 28, 2014 Expanded Revised Version: October 30, 2014

2 2 / 50 Outline I Math Model I Geometry I Test Data I Thermal Network Model Solution with TNSolver I Results

3 3 / 50 What Is New, By Revision Date October 30, Added geometry for 6.22 round copper fin 2. Added geometry for square copper fin 3. Added network diagram for 6.22 fin model 4. Added notes for running the supplied model files

4 4 / 50 The Control Volume Concept Math Model Energy In Energy Out = Energy Stored, Generated and/or Consumed Heat (transfer) is thermal energy transfer due to a temperature difference ˆn A da V

5 5 / 50 Convection Correlations Math Model The heat flow rate is: Q = ha(t s T ) where h is the heat transfer coefficient, T s is the surface temperature and T is the fluid temperature

6 External Forced Convection over a Cylinder Math Model Equation (7.44), page 436 in [BLID11] Nu D hd k = CRem 1/3 D Pr where D is the diameter of the cylinder and the Reynolds number is Re D = ρvd/µ = VD/ν. Re D C m , ,000 40, , , Table 7.2, page 437 in [BLID11] 6 / 50

7 7 / 50 External Forced Convection over a Noncircular Cylinder Math Model For the case of a gas flowing over noncircular cylinders in crossflow (Table 7.3, page 437 in [BLID11]): Geometry Re D C m p D = p 2H 2 6,000 60, g D = H 5,000 60, Note that the fluid properties are evaluated at the film temperature, T f : T f = T s + T 2

8 8 / 50 External Natural Convection over a Horizontal Cylinder Math Model Equation (9.34), page 581 in [BLID11]: Nu D = hd k = Ra 1/6 D [1 + (0.559/Pr) 9/16] 8/27 valid for Ra D 10 12, Pr 0.7, where D is the diameter of the cylinder and the Rayleigh number, Ra, is: Ra D = GrPr = gρ2 cβd 3 (T s T ) kµ = gβd3 (T s T ) να Note that the fluid properties are evaluated at the film temperature, T f : T f = T s + T 2 2

9 9 / 50 Surface Radiation Math Model Radiation exchange between a surface and large surroundings The heat flow rate is (Equation (1.7), page 10 in [BLID11]): Q = σɛ s A s (T 4 s T 4 sur ) where σ is the Stefan-Boltzmann constant, ɛ s is the surface emissivity and A s is the area of the surface. Note that the surface area, A s, must be much smaller than the surrounding surface area, A sur : A s A sur Note that the temperatures must be the absolute temperature, K or R

10 10 / 50 Radiation Heat Transfer Coefficient Math Model Define the radiation heat transfer coefficient, h r (see Equation (1.9), page 10 in [BLID11]): Then, Note: h r = ɛσ(t s + T sur )(T 2 s + T 2 sur ) Q = h r A s (T s T sur ) I h r is temperature dependent I h r can be used to compare the radiation to the convection heat transfer from a surface, h (if T sur and T have similar values)

11 h r (W/m 2 -K) 11 / 50 Range of Radiation Heat Transfer Coefficient Math Model Radiation Heat Transfer Coefficient, h r, for T 1 = 25 C s = s = s = s = " T = T s - T 1 (C)

12 Fin Convection Experiment Geometry 12 / 50

13 Fins to be Studied Geometry I Four uniform cross section fins will be studied I Three are circular,, and one is diamond shaped p (a square, rotated 45 ) Fin Material Dimensions k (W/m-K) Aluminum Alloy 6061-T6 0.5 X L Copper Alloy X L Copper Alloy X 6.77 L Stainless Steel Alloy 0.5 X L / 50

14 Thermocouple Locations Geometry Position Along the m Fin (m) Base TC1 TC2 TC3 TC4 TC5 TC / 50

15 15 / 50 Surface Radiation Properties Geometry Fin Material Emissivity, ɛ (dimensionless) Aluminum Alloy 6061-T Copper Alloy Stainless Steel Alloy See Table A.8, page 922, in [BLID11] See Table 10.1, page 530, in [LL12]

16 16 / 50 Measurements Test Data The following measurements are made: I Base temperature of the fin I Heater power I Ambient air temperature I Fin temperatures along the length of the fin I Thermocouples are located at the center line of the fin I Flow velocity for forced convection A typical data set is shown for the aluminum fin, for both forced and natural convection.

17 Temperature (C) 17 / 50 Aluminum Fin - Forced Convection Test Data Heater Power = W Aluminum Fin - Forced Convection Base = (C) TC1 = (C) TC2 = (C) TC3 = (C) TC4 = (C) TC5 = (C) TC6 = (C) Ambient = (C) Time

18 Temperature (C) 18 / 50 Aluminum Fin - Natural Convection Test Data Heater Power = W Aluminum Fin - Natural Convection Base = (C) TC1 = (C) TC2 = (C) TC3 = (C) TC4 = (C) TC5 = (C) TC6 = (C) Ambient = (C) Time

19 19 / 50 Heat Transfer Analysis Thermal Network Model I Energy conservation: control volumes I Identify and sketch out the control volumes I Use the conductor analogy to represent energy transfer between the control volumes and energy generation or storage I Conduction, convection, radiation, other? I Capacitance I Sources or sinks I State assumptions and determine appropriate parameters for each conductor I Geometry, material properties, etc. I Which conductor(s)/source(s)/capacitance(s) are important to the required results? I Sensitivity analysis I What is missing from the model? - peer/expert review

20 20 / 50 Thermal Network Terminology Thermal Network Model I Geometry I Control Volume I Volume property, V = dv V I Node:, T node = T (x V i)dv I Control Volume Surface I Area property, A = da A I Surface Node:, T surface node = T (x A i)da I Material properties I Conductors I Conduction, convection, radiation I Boundary conditions I Boundary node:

21 21 / 50 Thermal Network Analysis Thermal Network Model For the long fin, there are six thermocouples: Tc base tip 127 Tc Tr Tr

22 22 / 50 Thermal Network Analysis Thermal Network Model For the 6.77 long fin, there are four thermocouples: Tc base tip 125 Tc Tr 221 Tr

23 23 / 50 Control Volume Geometry Thermal Network Model Scale drawing of thermocouple locations and control volume lengths: base tip Fin Dimensions (m) base tip Fin Dimensions (m) Note that the control volume boundaries are located midway between the thermocouple locations

24 Round Fin, Long, Conduction Geometry Thermal Network Model Axial conduction in the fin, L = = m D = 0.5 = m and r = 0.25 = m: L = distance between TC A = πr 2 Conductor Length (m) A (m 2 ) / 50

25 Round Fin, Long, Convection Geometry Thermal Network Model Surface convection area: D = 0.5 = m A = L CV P = L CV πd Conductor L CV (m) A (m 2 ) / 50

26 Round Fin, 6.77 Long, Conduction Geometry Thermal Network Model Axial conduction in the fin, L = 6.77 = m D = 0.5 = m and r = 0.25 = m: L = distance between TC A = πr 2 Conductor Length (m) A (m 2 ) / 50

27 Round Fin, 6.77 Long, Convection Geometry Thermal Network Model Surface convection area: D = 0.5 = m A = L CV P = L CV πd Conductor L CV (m) A (m 2 ) / 50

28 Square Fin, Long, Conduction Geometry Thermal Network Model Axial conduction in the fin, L = = m, H = 0.5 = m L = distance between TC A = H 2 Conductor Length (m) A (m 2 ) / 50

29 Square Fin, Long, Convection Geometry Thermal Network Model Surface convection area: H = 0.5 = m A = L CV P = L CV 4H Conductor L CV (m) A (m 2 ) / 50

30 30 / 50 Conduction Conductor Thermal Network Model Q ij = ka L (T i T j ) Begin Conductors! label type nd_i nd_j k L A 100 conduction base 1 x.x conduction 1 2 x.x conduction 2 3 x.x conduction 3 4 x.x conduction 4 5 x.x conduction 5 6 x.x conduction 6 tip x.x End Conductors

31 31 / 50 Convection Conductor Thermal Network Model Q ij = ha(t s T ) = ha(t i T j ) Begin Conductors! label type nd_i nd_j h A 121 convection 1 Tc x.x convection 2 Tc x.x convection 3 Tc x.x convection 4 Tc x.x convection 5 Tc x.x convection 6 Tc x.x convection tip Tc x.x End Conductors

33 33 / 50 Forced Convection Over Diamond Conductor Thermal Network Model Heat transfer coefficient, h, is evaluated using the correlation. Begin Conductors! label type nd_i nd_j fluid V D A 121 EFCdiamond 1 Tc air x.x EFCdiamond 2 Tc air x.x EFCdiamond 3 Tc air x.x EFCdiamond 4 Tc air x.x EFCdiamond 5 Tc air x.x EFCdiamond 6 Tc air x.x EFCdiamond tip Tc air x.x End Conductors Note that Re, Nu and h are reported in the output file.

34 34 / 50 Natural Convection Conductor Thermal Network Model Heat transfer coefficient, h, is evaluated using the correlation. Begin Conductors! label type nd_i nd_j fluid D A 121 ENChcyl 1 Tc air ENChcyl 2 Tc air ENChcyl 3 Tc air ENChcyl 4 Tc air ENChcyl 5 Tc air ENChcyl 6 Tc air ENChcyl tip Tc air End Conductors Note that Ra, Nu and h are reported in the output file.

35 35 / 50 Surface Radiation Conductor Thermal Network Model Q ij = σɛ i A i (T 4 i T 4 j ) Begin Conductors! label type nd_i nd_j emissivity A 221 surfrad 1 Tr x.xx surfrad 2 Tr x.xx surfrad 3 Tr x.xx surfrad 4 Tr x.xx surfrad 5 Tr x.xx surfrad 6 Tr x.xx surfrad tip Tr x.xx End Conductors Note that h r is reported in the output file.

36 36 / 50 Boundary Conditions Thermal Network Model Begin Boundary Conditions! type Tb Node(s) fixed_t x.xx Tc! fluid T fixed_t x.xx Tr! surrounding radiation T fixed_t x.xx base! fin base T End Boundary Conditions

37 TNSolver Input Files Thermal Network Model The supplied TNSolver input files for each fin model: Fin Forced Convection Natural Convection Aluminum, Al fin FC.inp Al fin NC.inp Round Copper, 6.77 Cu fin FC.inp Cu fin NC.inp Square Copper, sqcu fin FC.inp sqcu fin NC.inp Stainless Steel, SS fin FC.inp SS fin NC.inp Edit each file to set appropriate boundary conditions (Tc, Tr and base) and flow velocities (forced convection models) for your specific experimental data set 37 / 50

38 38 / 50 Running a TNSolver Model Thermal Network Model The MATLAB/Octave command to run the aluminum fin, forced convection model is: >>tnsolver( Al fin FC ); Note that you supply the base of the input file name (AL fin FC, in this example), without the.inp extension The results output file will be the base name with the extension.out (Al fin FC.out, in this example)

39 39 / 50 Screen Output When Running TNSolver Thermal Network Model ********************************************************** * * * TNSolver - A Thermal Network Solver * * * * Version 0.2.0, October 28, 2014 * * * ********************************************************** Reading the input file: Al_fin_FC.inp Initializing the thermal network model... Starting solution of a steady thermal network model... Nonlinear Solve Iteration Residual e e-09 Results have been written to: Al_fin_FC.out All done...

40 40 / 50 View TNSolver Output File For Results Thermal Network Model Open Al fin FC.out in your favorite text editor ********************************************************** * * * TNSolver - A Thermal Network Solver * * * * Version 0.2.0, October 28, 2014 * * * ********************************************************** Model run finished at 10:05 AM, on October 29, 2014 *** Solution Parameters *** Title: Aluminum Fin - Forced Convection Type = steady Units = SI Temperature units = C Nonlinear convergence = 1e-008 Maximum nonlinear iterations = 15 Gravity = (m/sˆ2) Stefan-Boltzmann constant = e-008 (W/mˆ2-Kˆ4)

41 41 / 50 View TNSolver Output File For Results (continued) Thermal Network Model *** Nodes *** Volume Temperature Label Material (mˆ3) (C) N/A N/A N/A N/A N/A N/A Tc N/A 0 25 Tr N/A 0 25 base N/A tip N/A

42 42 / 50 View TNSolver Output File For Results (continued) Thermal Network Model *** Conductors *** Q_ij Label Type Node i Node j (W) conduction base conduction conduction conduction conduction conduction conduction 6 tip EFCcyl 1 Tc EFCcyl 2 Tc EFCcyl 3 Tc EFCcyl 4 Tc EFCcyl 5 Tc EFCcyl 6 Tc EFCcyl tip Tc surfrad 1 Tr surfrad 2 Tr surfrad 3 Tr surfrad 4 Tr surfrad 5 Tr surfrad 6 Tr surfrad tip Tr

43 43 / 50 View TNSolver Output File For Results (continued) Thermal Network Model *** Conductor Parameters *** surfrad: Surface Radiation h_r label (W/mˆ2-K) EFCcyl: External Forced Convection - Cylinder h label Re Number Nu Number (W/mˆ2-K)

44 44 / 50 Verification with Analytical Solution Results Table 3.4, page 161 in [BLID11], Temperature distribution and heat loss for fins with a uniform cross section Case Tip Condition Temperature, θ/θ b Fin Heat Transfer Rate, Q A Convection cosh m(l x)+(h/mk) sinh m(l x) sinh ml+(h/mk) cosh ml cosh ml+(h/mk) sinh ml M cosh ml+(h/mk) sinh ml cosh m(l x) cosh ml M tanh ml (θ L /θ b ) sinh mx+sinh m(l x) sinh ml M cosh ml (θ L/θ b ) sinh ml B Adiabatic C Specified T D L = 1 e mx M θ = T T m = hp/ka c θ b = θ(0) = T b T M = ( p hpka c )θ b For circular fins: P = πd and A c = π(d/2) 2 For square/diamond fins: P = 4H and A c = H 2

45 Temperature (C) 45 / 50 Comparison with TNSolver Network Model Results 50 1/2" Diameter Fin, Case A Tip, h = 55.0 W/m2 -K SS, Q = W Aluminum, Q = W Copper, Q = W TNSolver SS, Q = 1.14 W TNSolver Al, Q = 5.1 W TNSolver Cu, Q = 7.7 W x (m)

46 TNSolver Forced Convection Correlation Results Aluminum fin, flow velocity = 3.7 m/s = 8.3 mph T amb = 25.0C and T base = 55.8C Experiment TNSolver Q c Q r h h r TC (C) (C) (W) (W) (W /m 2 K ) (W /m 2 K ) tip Q f = 6.1 W Re range: 2,833 2,981 Nu range: / 50

47 47 / 50 TNSolver Natural Convection Correlation Results Aluminum fin T amb = 26.3 and T base = 86.1C Experiment TNSolver Q c Q r h h r TC (C) (C) (W) (W) (W /m 2 K ) (W /m 2 K ) tip Q f = 2.8 W Ra range: 6,000 7,300 Nu range:

48 residual 48 / 50 Al Fin Forced Convection Least Squares Fit h Results A constant heat transfer coefficient, h, was determined using a least squares residual between the experimental data and the TNSolver thermal network model. h = 49.3 W /m 2 K Experiment TNSolver TC T (C) T (C) Best Fit h = h

49 residual 49 / 50 Al Fin Natural Convection Least Squares Fit h Results A constant heat transfer coefficient, h, was determined using a least squares residual between the experimental data and the TNSolver thermal network model. h = 10.3 W /m 2 K Experiment TNSolver TC T (C) T (C) Best Fit h = h

50 50 / 50 Conclusion I TNSolver thermal network model verification with analytical solution demonstrated validity of the model I The forced convection correlation was within experimental error I The natural convection correlation was within 20% of the experimental temperature data I The forced convection heat transfer coefficient determined using least squares fitting was within 10% of the correlation I The natural convection heat transfer coefficient determined using least squares fitting was 2.3 times the correlation h Questions?

51 51 / 50 Text Editors The input file to TNSolver is a plain text file It is not a good idea to use a word processor to edit the files Some recommendations: I Cross-platform: vim/gvim, emacs, Bluefish, among many others I Windows: notepad, Notepad++ I MacOS: TextEdit, Smultron I Linux: see cross-platform options

52 52 / 50 References I [BLID11] T.L. Bergman, A.S. Lavine, F.P. Incropera, and D.P. DeWitt. Introduction to Heat Transfer. John Wiley & Sons, New York, sixth edition, [LL12] J. H. Lienhard, IV and J. H. Lienhard, V. A Heat Transfer Textbook. Phlogiston Press, Cambridge, Massachusetts, fourth edition, Available at:

Fin Convection Experiment

Fin Convection Experiment Thermal Network Solution with TNSolver Bob Cochran Applied Computational Heat Transfer Seattle, WA TNSolver@heattransfer.org ME 331 Introduction to Heat Transfer University of