Professor Faith Morrison

Transcription

1 CM3215 Fundamentals of Chemical Engineering Laboratory Professor Faith Morrison Department of Chemical Engineering Michigan Technological University 1 Objective: Measure the heat-transfer coefficient to a 1.0 brass sphere dropped into in a well-stirred beaker of hot water. Strategy:? 2 1

2 Objective: Measure the heat-transfer coefficient to a 1.0 brass sphere dropped into in a well-stirred beaker of water. Strategy:? To determine our experimental strategy, we begin with: What is heat transfer coefficient? Newton s Law of Cooling: (a boundary condition) Perhaps: measure and? 3 How to experimentally measure heat-transfer coefficient? Newton s Law of Cooling: (a boundary condition) Perhaps: measure and? 4 2

3 How to experimentally measure heat-transfer coefficient? Newton s Law of Cooling: (a boundary condition) Perhaps: measure and? But they will be equal! Need heat transfer to be taking place. 5 How to experimentally measure heat-transfer coefficient? Newton s Law of Cooling: (a boundary condition) Perhaps: measure and? But they will be equal! Need heat transfer to be taking place. Unsteady state experiment, then. It would be awkward to try to measure the surface temperature. 6 3

4 How to experimentally measure heat-transfer coefficient? Newton s Law of Cooling: (a boundary condition) Perhaps: measure and? But they will be equal! Need heat transfer to be taking place. Unsteady state experiment, then. It would be awkward to try to measure the surface temperature. We can embed a thermocouple in the center perhaps? How is the temperature at the center related to what we seek to measure,? 7 How to experimentally measure heat-transfer coefficient? Newton s Law of Cooling: (a boundary condition) Perhaps: measure and? But they will be equal! Need heat transfer to be taking place. Unsteady state experiment, then. It would be awkward to try to measure the surface temperature. We can embed a thermocouple in the center perhaps? How is the temperature at the center related to what we seek to measure,? We can model the situation and see what the connection is. 8 4

5 Experiment: Measure T(t) at the center of a sphere: Initially: T-couple measures T(t) at the center of the sphere 9 Experiment: Measure T(t) at the center of a sphere: Initially: Suddenly: T-couple measures T(t) at the center of the sphere Heat transfer takes place. 10 5

6 Experiment: Measure T(t) at the center of a sphere: Initially: Suddenly: (already done for you) T-couple measures T(t) at the center of the sphere Excel: t(s) T( o C) 9.50E E E E E E E E E E E E E E E E E E E E E E E E E E Experiment: Measure T(t) at the center of a sphere: How do we set up the model for this problem? Where does h come into it? Modeling exercise: How does the temperature at the center of a sphere, subjected to this history, change with time? 12 6

7 Microscopic Energy Balance 13 Modeling exercise: How does the temperature at the center of a sphere, subjected to this history, change with time? You try

8 Unsteady State Heat Transfer to a Sphere Microscopic energy balance in the sphere: Boundary conditions:, 0, Initial condition: 0, 1 0 Unsteady Solid 0, symmetry No current, no rxn 0 15 Unsteady State Heat Transfer to a Sphere Microscopic energy balance in the sphere: Boundary conditions:, 0, Initial condition: 0, 1 0 Unsteady Solid 0, symmetry No current, no rxn 0 ( means for all ) 16 8

9 Unsteady State Heat Transfer to a Sphere Microscopic energy balance in the sphere: 1 Unsteady Solid 0, symmetry Now, Solve No current, no rxn Boundary conditions:, 0 0, 0 Initial condition: 0, ( means for all ) 17 Unsteady State Heat Transfer to a Sphere Microscopic energy balance in the sphere: Boundary conditions:, 0, Initial condition: 0, Unsteady Solid 0, symmetry No current, no rxn 0 ( means for all ) 18 9

10 Unsteady State Heat Transfer to a Sphere Solution:,, 2Bi (Bi=Biot number; Fo=Fourier number) Bi sin Fo sin Bi 1 BiBi1 where the eigenvalues satisfy this equation: tan Bi10 Characteristic Equation (Carslaw and Yeager, 1959, p237) 19 Unsteady State Heat Transfer to a Sphere Solution: Depends on material ( /, and heat transfer processes at surface 2Bi (Bi=Biot number; Fo=Fourier number),, Bi sin Fo sin Bi 1 BiBi1 where the eigenvalues satisfy this equation: tan Bi10 Characteristic Equation (Carslaw and Yeager, 1959, p237) 20 10

11 Unsteady State Heat Transfer to a Sphere Solution: What does this look like? Fo Bi 2Bi sin sin Bi 1 BiBi1 where the eigenvalues satisfy this equation: Let s plot it to find Bi10 tan Characteristic Equation 21 out. (Excel) Let s plot it to find out: what are the variables? Solution: Bi Fo 2Bi sin sin Bi 1 BiBi1 2Bi Exponential decay with Fo (scaled time) bunch of terms that vary with Bi and Bi Bi varies with Bi: tan Bi 10 Characteristic Equation If we choose a fixed Bi, then only Fo varies 22 11

12 Unsteady State Heat Transfer to a Sphere If we choose a fixed Bi, then only Fo varies 2Bi For a fixed Bi Bi,, term n=1 (sum of 9 terms) term n= term n=2 Biot number= first term second term third term fourth term fifth term sixth term seventh term eighth term ninth term Fo fixed, R, k 23 Unsteady State Heat Transfer to a Sphere If we fix Bi, then only Fo varies:,, term n=1 (sum of 9 terms) term n= term n=2 It is an infinite sum of terms. Each term corresponds to one eigenvalue, The first term 1is the dominant one The 1 terms alternate in sign (positive and negative) Higher terms are fixing the short time behavior Biot number= first term second term third term fourth term fifth term sixth term seventh term eighth term ninth term Fo For a fixed Bi Bi 24 12

13 Unsteady State Heat Transfer to a Sphere If we fix Bi, then only Fo varies:,, result (9 terms) Biot number= first term second term For 0.2, the third term higher-order terms fourth term fifth term make no contribution sixth term seventh term eighth term ninth term Fo h.o.t Unsteady State Heat Transfer to a Sphere If we fix Bi, then only Fo varies:,, result (9 terms) Biot number= first term second term For 0.2, the third term higher-order terms fourth term fifth term make no contribution sixth term seventh term eighth term ninth term The higher-order terms are working to get the solution right Fo for shorter and shorter times (low Fo) h.o.t

14 Unsteady State Heat Transfer to a Sphere If we fix Bi, then only Fo varies:,, result (9 terms) We already know the short-time behavior; we thus concentrate 0.2 on long times to 0.4fit the data (and therefore need only one 0.6 term of the summation) Biot number= first term second term For 0.2, the third term higher-order terms fourth term fifth term make no contribution sixth term seventh term eighth term ninth term The higher-order terms are working to get the solution right Fo for shorter and shorter times (low Fo) h.o.t. 27 Unsteady State Heat Transfer to a Sphere If we fix Bi, then only Fo varies:,, Scaled Temperature Fo 0.2 nine terms first term Plotted log-linear, the solution is linear for 0.2 The slope depends on Biot number Fourier number Fo Bi 1 Bi 28 14

15 Unsteady State Heat Transfer to a Sphere If we fix Bi, then only Fo varies: (summary) For a fixed Bi: Bi the results are only a function of Fo. Fo Solution is an infinite sum of terms. Each term corresponds to one eigenvalue, The first term 1is the dominant one The 1 terms alternate in sign (positive and negative) Higher terms are fixing the short time behavior At fixed Biot number, the time-dependence is an exponential decay (for Fo 0.2 Our Biot number will be fixed; but we not know what it is until we fit our results to the model. Question: How do various values of Biot number affect the heat transfer that occurs? 29 Unsteady State Heat Transfer to a Sphere The roots of the characteristic equation vary with Bi Characteristic Equation: 14.0 tan Bi10 R n low Bi is high k, low h Biot Number high high Bi Bi is low is conductivity low k, high h first mode second mode third mode fourth mode The are the roots of the characteristic equation They depend on Biot number Bi Bi 30 15

16 Unsteady State Heat Transfer to a Sphere The roots of the characteristic equation vary with Bi Characteristic Equation: 14.0 Low Bi: high, low tan Bi10 R n low Bi is high k, low h Biot Number first mode second mode third mode fourth mode At low Bi, the high high Bi Bi low is temperature conductivity is uniform low k, high h in the sphere; heat transfer is limited by rate of heat transfer to the surface (). The are the roots of the characteristic equation They depend on Biot number Bi Bi 31 Unsteady State Heat Transfer to a Sphere the 14.0 bulk temperature; R n The roots of the characteristic equation vary with Bi Characteristic Equation: At high Bi, the surface temperature equals heat transfer is limited 12.0 by conduction in the 10.0 sphere low Bi is high k, low h Biot Number high high Bi Bi is low is conductivity low k, high h High Bi: low, high first mode second mode third mode fourth mode tan Bi10 The are the roots of the characteristic equation They depend on Biot number Bi Bi 32 16

17 Initially: Suddenly: Heat Xfer Coeff for Sphere Assignment 8 4/4/2016 At moderate Bi, heat transfer is affected Unsteady by State both Heat conduction Transfer in to a Sphere the The sphere roots of the and characteristic the rate equation of heat vary transfer with Bi to the surface. Characteristic Equation: R n low Bi is high k, low h Biot Number high high Bi Bi is low is conductivity low k, high h Moderate Bi: nether process dominates first mode second mode third mode fourth mode tan Bi10 The are the roots of the characteristic equation They depend on Biot number Bi Bi 33 The key to analyzing our data is determining the Biot number for the experiment. We determine the Biot number by fitting the model predictions to our data. The value of Biot number that results in the closest fit between model and data tells us our value of. Bi Scaled Temperature Fourier number nine terms first term R n Biot Number high Bi is low conductivity first mode second mode third mode fourth mode 34 17

18 Summary: The key to analyzing our data is determining the Biot number for the data. We determine the Biot number by fitting the model predictions to our data. The value of Biot number that results in the closest fit between model and data tells us our value of. In the literature, there are handy plots of the solutions to Unsteady State Heat Transfer to a Sphere (and other shapes) Heisler charts (Geankoplis; see also Wikipedia) From Geankpolis, 4 th edition, page Literature solutions to Unsteady State Heat Transfer to a Sphere Heisler charts (Geankoplis; see also Wikipedia) From Geankpolis, 4 th edition, page

19 Literature solutions to Unsteady State Heat Transfer to a Sphere Heisler charts (Geankoplis; see also Wikipedia) The Heisler Chart is a catalog of all the Y long-time shapes for various values of Biot number Bi. From Geankpolis, 4 th edition, page Literature solutions to Unsteady State Heat Transfer to a Sphere Heisler charts (Geankoplis; see also Wikipedia) The Heisler Chart is a catalog of all the Y long-time shapes for various values of Biot number Bi. Our data correspond to only one of these lines; when we know which line, we know Bi. From Geankpolis, 4 th edition, page

20 Literature solutions to Unsteady State Heat Transfer to a Sphere Heisler charts (Geankoplis; see also Wikipedia) All the lines go through the point, 0,1 The Heisler Chart is a catalog of all the Y long-time shapes for various values of Biot number Bi. Our data correspond to only one of these lines; when we know which line, we know Bi. From Geankpolis, 4 th edition, page Literature solutions to Unsteady State Heat Transfer to a Sphere Heisler charts (Geankoplis; see also Wikipedia) Note: the parameter m from the Geankoplis Heisler chart is NOT the slope of the line! It is a label of Bi. 1 Bi Also, x 1 is the sphere radius From Geankpolis, 4 th edition, page

21 Heisler charts (Geankoplis; see also Wikipedia) Note also: think of it as 4 separate graphs 10 From Geankpolis, 4 th edition, page Assignment 8: Unsteady Heat Transfer to a Sphere (team assignment) Using the time-dependent temperature versus time data measured by colleagues (supplied to you), calculate the heat transfer coefficient for the laboratory experiment (sphere plunged into a beaker of hot water). Report the Biot number. Report the heat transfer coefficient,. Indicate which physics is dominating the dynamics: heat transfer to the sphere (), heat transfer within the sphere (), or neither dominate. Describe your method for obtaining Bi and (briefly; you may give steps as a list in your memo; it s not a report). Conduct and report an uncertainty analysis to determine error limits on

22 Which physics dominates? At low Bi, the temperature is uniform in the sphere; heat transfer is limited by rate of heat transfer to the surface (). At high Bi, the surface temperature equals the bulk temperature; heat transfer is limited by conduction in the sphere (). At moderate Bi, heat transfer is affected by both conduction in the sphere and the rate of heat transfer to the surface (both and influence the heat transfer). The key to analyzing our data is determining the Biot number for the experiment. 43 Pre-lab Assignment (due Tuesday in lab): In your lab notebook, outline how you plan to proceed; show the TA. Obtain / and for brass ahead of time and record both in notebook (with reference) Submit your data to the TA for the archive: DP meter calibration Orifice meter calibration (two different units) Rotameter calibration Lossy pump characteristic curve 44 22