heat xfer coeff for sphere assignment 8fmorriso/cm3215/lectures/cm3215_lecture10heat... · heat...

Heat Xfer Coeff for Sphere Assignment 8

1

CM3215

Fundamentals of Chemical Engineering Laboratory

Professor Faith Morrison

Department of Chemical EngineeringMichigan Technological University

© Faith A. Morrison, Michigan Tech U.

1

CM3215 Fundamentals of Chemical Engineering Laboratory

In the previous lab, we practiced

curve-fitting data to an empirical model

In this lab, we use this skill to deduce a quantity that is meaningful to us

from an analytical model:

Heat transfer coefficient, h


• It’s convenient to have a functional form for future use

• The parameters may have no meaning

• Model an actual experiment (physics)

• Goal is to determine a quantity that appears as a parameter in the model

• Fit data to model to determine the parameter

2


2


Question: What is the heat-transfer coefficient to a small object in a particular well stirred vessel?

Answer: We will measure heat flow and back-out h from modeling our data.


Newton’s Law of Cooling:

(a boundary condition)

(What is heat transfer coefficient? Answer: go back to the definition

3



Experiment: Measure T(t) for an unsteady state scenario:

4

Initially:


3




5

Initially: Suddenly:




6

Initially: Suddenly:

t(s) T(oC)9.50E-02 7.46E+002.11E-01 7.44E+003.09E-01 7.44E+004.09E-01 7.57E+005.24E-01 7.46E+006.23E-01 7.49E+007.39E-01 7.53E+008.37E-01 7.46E+009.54E-01 7.59E+001.05E+00 7.53E+001.15E+00 7.58E+001.27E+00 7.48E+001.37E+00 7.57E+00… …

Excel:

(already done for you)


4



How do we set up the model for this problem?

Where does h come into it?

7


www.chem.mtu.edu/~fmorriso/cm310/energy2013.pdf8

Microscopic Energy Balance


5


www.chem.mtu.edu/~fmorriso/cm310/energy2013.pdf9

You try.

© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.

Unsteady State Heat Transfer to a Sphere

1≡

Boundary conditions:

,

00,

Microscopic energy balance in the sphere:

Initial condition:

10

0, ∀

∀

0

(“∀” means “for all”)

• Unsteady

• Solid 0

• , symmetry

• No current, no rxn


6

© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.


1≡

Boundary conditions:

,

00,

Microscopic energy balance in the sphere:

Initial condition:

11

0, ∀

∀

0

(“∀” means “for all”)

• Unsteady

• Solid 0

• , symmetry

• No current, no rxnNow, Solve



Solution:

(Carslaw and Yeager, 1959, p237)

, ≡,

Bi ≡

2Bisin sin Bi 1

Bi Bi 1

Fo ≡

12

where the eigenvalues satisfy this equation:tan

Bi 1 0

(Bi=Biot number; Fo=Fourier number)

Characteristic Equation


7

Depends on material ( and heat transfer at

surface

2Bisin sin Bi 1

Bi Bi 1



Solution:

(Carslaw and Yeager, 1959, p237)

, ≡,

Bi ≡ Fo ≡

13


Bi 1 0

(Bi=Biot number; Fo=Fourier number)


Let’s plot it to find out. © Faith A. Morrison, Michigan Tech U.


Solution: ≡

Bi ≡

2Bisin sin Bi 1

Bi Bi 1

Fo ≡

14


Bi 1 0

But what does this look like?



8

2Bi

sin sin Bi 1Bi Bi 1

‐0.6

‐0.4

‐0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.1 0.2 0.3 0.4

Biot number= 1.0000

first term

second term

third term

fourth term

fifth term

sixth term

seventh term

eighth term

ninth term

© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.


15

Fo ≡,≡

,

result (9 terms)

Bi ≡

For a fixed Bi

‐0.6

‐0.4

‐0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.1 0.2 0.3 0.4

Biot number= 1.0000

first term

second term

third term

fourth term

fifth term

sixth term

seventh term

eighth term

ninth term

© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.


16

Fo ≡,≡

,

result (9 terms)

•It is an infinite sum of terms.

•Each term corresponds to one eigenvalue,

•The first term is the biggest

•The terms alternate in sign (positive and negative)

Bi ≡

For a fixed Bi


9



17

‐20

‐10

0

10

20

30

40

0 2 4 6 8 10 12 14 16

characteristic equation

Rn

Biot number= 1.0000

Characteristic Equation:tan

Bi 1 0

• The are the roots of the characteristic equation

• There is a different set of for each value of Biot number Bi

Bi ≡

For Bi=1, 1.5, 4.6



18

Characteristic Equation:

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.001 0.1 10 1000

R

n

Biot Number

first mode

second mode

third mode

fourth mode

high Bi is low conductivity

low Bi is high k, low h

high Bi islow k, high h


• They depend on Biot number Bi

Bi ≡

The roots of the characteristic equation vary with Bi

tanBi 1 0


10



19


0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.001 0.1 10 1000

R

n

Biot Number

first mode

second mode

third mode

fourth mode






Bi ≡


tanBi 1 0

At low Bi, the temperature is uniform

in the sphere; heat transfer is limited by

rate of heat transfer to the surface (h).



20


0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.001 0.1 10 1000

R

n

Biot Number

first mode

second mode

third mode

fourth mode






Bi ≡


tanBi 1 0

At high Bi, the surface temperature equals

the bulk temperature; heat transfer is limited by conduction in the

sphere.


11



21


0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.001 0.1 10 1000

R

n

Biot Number

first mode

second mode

third mode

fourth mode






Bi ≡


tanBi 1 0

At moderate Bi, heat transfer is affected by both conduction in the sphere and the rate of heat

transfer to the surface.

2Bi

sin sin Bi 1Bi Bi 1

‐0.6

‐0.4

‐0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.1 0.2 0.3 0.4

Biot number= 1.0000

first term

second term

third term

fourth term

fifth term

sixth term

seventh term

eighth term

ninth term

© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.


22

Fo ≡,≡

,

result (9 terms)

The key to analyzing data is determining

the Biot number.


12

‐0.6

‐0.4

‐0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.1 0.2 0.3 0.4

Biot number= 1.0000

first term

second term

third term

fourth term

fifth term

sixth term

seventh term

eighth term

ninth term



23

Fo ≡

,≡

,

result (9 terms)

For 0.2, the higher-order terms

make no contribution

‐0.6

‐0.4

‐0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.1 0.2 0.3 0.4

Biot number= 1.0000

first term

second term

third term

fourth term

fifth term

sixth term

seventh term

eighth term

ninth term



24

Fo ≡

,≡

,

result (9 terms)



The higher-order terms are working to get the solution right

for shorter and shorter times (low Fo)


13

‐0.6

‐0.4

‐0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.1 0.2 0.3 0.4

Biot number= 1.0000

first term

second term

third term

fourth term

fifth term

sixth term

seventh term

eighth term

ninth term



25

Fo ≡

,≡

,

result (9 terms)



The higher-order terms are working to get the solution right

for shorter and shorter times (low Fo)

We already know the short-time behavior; we

thus concentrate on long times to get Bi (and

therefore need only one term of the summation)



26

0.001

0.01

0.1

1

10

0.0 0.2 0.4 0.6 0.8 1.0

Scaled Temperature

Fourier number

nineterms

firstterm

,≡

,

Fo ≡

• Plotted log-linear, the solution is linear for 0.2

• The slope depends on Biotnumber

Bi ≡

1Bi


14

From Geankpolis, 4th edition, page 374

© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.

27

Heisler charts (Geankoplis; see also Wikipedia)Solution to Unsteady State Heat Transfer to a Sphere


© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.

28


The Heisler Chart is a catalog of all the long-time slopes for various values of Biot number.


15


© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.

29


The Heisler Chart is a catalog of all the long-time slopes for various values of Biot number.

Our data correspond to only one of these lines; when we know which

one, we know Bi.



© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.

Also, x1 is the sphere radius

Note: the parameter m from the Geankoplis Heisler chart is NOT the

slope of the line! It is Bi.

30

1Bi


16


Assignment 8: Unsteady Heat Transfer to a Sphere

•Using the time-dependent temperature versus time data measured by others, calculate the heat transfer coefficient for the conditions in the laboratory experiment (sphere in a beaker).

•Report the parameters of the fit.

•Demonstrate the “goodness of fit” with a plot of data (symbols) and model (solid line).

•Conduct and report your uncertainty analysis.


31


•In your lab notebook, outline how you plan to proceed

•Obtain / and for brass ahead of time and record both in notebook (with reference)


Pre-lab Assignment (due in lab Tuesday):

•Have your calibration curves in your lab notebook for collection by the TA

Reminder:

32

heat xfer coeff for sphere assignment 8fmorriso/cm3215/lectures/cm3215_lecture10heat... · heat...

Documents