The result of that problem was the following ODE: a) Using the Runge-Kutta techniques discussed in class, assume constant properties and calculate the value of T(t) for the iron base plate over 8 minutes from an initial temperature of 25oC assuming an electrically generated heat flux of q?=10000 W/m2 (as opposed to the 12500 W/m2 in the example). Use the following approaches:

Project HW2

Due ~11/9

The following problems are to be solved computationally using the methods that have been

presented in class. Direct ODE solvers may be used to check your work, but the presentation

of your results should include an explicit example and discussion of your numerical approach

including either a spreadsheet, a Matlab routine, a computer code, or similar evidence. The

presentation of your results should be in a typed format containing a brief description of your

work, summary of results and answers to additional questions, and any requested plots and

tables. In addition to what is requested by the individual problems, make clear in each case

what differential equation(s) you are solving in variable form. Note that you are welcome to

submit the code/Matlab/spreadsheet/etc information electronically (on a CD or via e-mail);

likewise, details of analytical work can be neatly handwritten; however, the final typed report

should contain all the critical results.

Each student is expected to complete 2 of the first 3 problems along with problem 4. You may

work in a group of up to 4 people on each problem. Note that they do not have to be the same

people on each problem ? Student A could work on problem 2 with students B & C, while

doing problem 3 with student D, and doing problem 4 solo for example. The grade for

student A would then be the summed score from problems 2, 3, and 4.You may not do the

remaining problem for extra credit, but you can obvioulsy do the extra credit associated with

the problems you do complete (Each problem has an extra credit. 1. (20 pts) Return to the problem in Example 20 from class. The result of that problem was the

following ODE:

4

4

dT q h A p (h[T T ] [T Tsur ]) A p dt

[( Vc) p ( Vc) t ] a) Using the Runge-Kutta techniques discussed in class, assume constant properties and

calculate the value of T(t) for the iron base plate over 8 minutes from an initial temperature

of 25oC assuming an electrically generated heat flux of q?=10000 W/m2 (as opposed to the

12500 W/m2 in the example). Use the following approaches:

1) 1st-order RK, Dt = 10 sec

2) 1st-order RK, Dt = 2 sec

3) 1st-order RK, Dt = 0.5 sec

4) 2nd-order RK Dt = 5 sec

5) 2nd-order RK, Dt = 1 sec

Plot all five of these results on a single T vs. t plot. Determine for each case the time needed

to achieve the desired temperature of 130oC and the time to reach 200oC. Compare these

results the solution one gets using the approach in Example L, accounting for the change in

the heat flux above.

b) New regulations require that the temperature of iron base plate cannot exceed 220oC.

Determine the electrical heat flux required to achieve this temperature in the steady-state case

by setting dT/dt = 0 in the ODE and solve for T. Using methods (2) and (5) above, determine

in this case how long it will take the iron to reach 130oC and 200oC.

Bonus Credit (5 pts)

The above solution assumes that the properties are temperature independent, which is not

actually the case. In addition, the convection coefficient can vary with temperature as

well. From the appendices in the book, obtain data for air at 1 atm and for the

aluminum alloy 2024. Use the average Nusselt equation for laminar flat plate flow: Nu L 0.664 Re1x/ 2 Pr1 / 3

to determine the average convection coefficient over the iron, assuming a freestream

velocity of 1 m/s and a plate length of 15 cm (the average width of the iron). Use the

film temperature to evaluate the properties of the Nusselt correlation. With these

variable quantities, redo part (a) above using either method (2) or (5) to determine the

time required to heat the iron to 130oC. Also determine the steady-state temperature

and the amount of time necessary to get within 10oC of that temperature. 2. (20 pts) The solution of the similarity equation for the laminar boundary layer: 2 d3 f

dh 3 f d2 f

dh 2 df

dh 0, f (0) 0,

h 0 df

dh 1

h Can also be solved as a multi-step initial value problem. In this case, three computations must

be done simultaneously: d2 f dh 2 d2 f 2 n 1 dh 3 (Dh ) d f dh 3

n d2 f df df (Dh ) dh 2 dh n 1 dh n 2 d f

2 n dh , n Originally, the goal was to guess at the value:

and integrate the equations to see if: df

dh 2 (Dh ) f d f 2 dh 2

n , n df f n 1 f n (Dh ) dh n

d2 f

dh 2

1 ?

h 0 h although infinity typically can be considered a value h>8. It is in this manner that the proper

value f?(0)=0.332 was determined. Since we already have that value, our goal will be to test

our methods.

a) do a 1st-order RK integration with Dh = 0.01 from h=0 to 8, using the proper values at h =

0. Reproduce the boundary layer equivalent to Table 7.1 using your data, but now showing

the three functions (f, f?, and f?) from 0 to 8 using a step of 0.2. What is the value of hhd

corresponding to the top of the thermal boundary layer (f?(hd) = 0.99) to two decimal places.

b) Redo (a), but with Dh = 0.002. How much does this change your result?

c) Instead of the proper value of f?(0), try values of 0.318, 0.330, 0.334, and 0.346. Redo (a)

with each of these initial guesses, creating new versions of Table 7.1. What is the resulting

percentage error in the value of hd relative to your result in part (a) for each case.

d) using the correct initial values as in (a), increase the value of Dh by increments of 0.01. At

what value of Dh does the error in the value hd become equal to or greater than the errors in

the three cases of (b) above? Is the computation more sensitive to the initial guess of f?(0) or

the size of Dh?

Bonus Credit (5 pts)

Solve the initial value problem of the thermal boundary layer equation:

d 2T * Pr dT * f 0, T * (0) 0, T * () 1

2

2

d

h

dh

using the same methods as (a) above for Pr =0.7. In this case, you will have to guess at the

initial value of dT*/dh, although the correlation in the text is a good starting point. Assume

the correct solution will be when T* = 1 for large h (h~20 usually is good). Get as close as

possible to that result, putting no more than three decimal places of accuracy on the initial

value of dT*/dh, and create the equivalent of Table 7.1 for the thermal boundary layer

equation, ranging over values of h=0 ? 20 and showing values for T* and dT*/dh at an

increment of Dh = 1. 3. (20 points) Examine our standard 1-D cooling problem with a wall. For Bi > 0.1, we can use

the 1D time-dependent plane wall solution (see chapter 5.5) to get a time-dependent series of

temperature profiles across the wall for symmetric cases. We will numerically compute the

results for 3 wall configurations, two symmetric cases and one not symmetric: L 10 cm

k 5 W/m K BC1 k, c p , a 4.0 10 -6 m 2 / s case i 3000 kg/m 3

at

Fot ( L / 2) 2 case ii BC 2 case iii BC1

Ts ,1 300 K

T 300 K BC 2

Ts , 2 300 K

T 300 K h 100 W/m2 K h 100 W/m2 K

Ts ,1 300 K T 400 K

h 100 W/m2 K L

a) For the three cases above, solve the time dependent problem numerically using a 1D

explicit approach (see text for 1D equations) assuming a uniform initial temperature of 500 K

in the wall for cases i, ii, and iii. Use 11 evenly-spaced points across the wall (one on each

wall, point 6 in the middle, Dx = L/10) in your computations. Compute through time Fot = 1.

Plot your profiles (T vs x) for Fot = 0,0.05,0.1,0.15,0.20,0.30,0.50, and 1.0 Clearly indicate

you choice of timestep. For case i and ii, compare your results to the exact solution approach

in 5.5.

b) Redo part (a) for case ii and iii using the implicit method. Try using a timestep of DFo =

0.1 as well as the timestep used in part (a). Compare your results at Fot =0,0.1,0.2,0.3,0.5,

and 1.0 to the results in part (a).

c) Determine the steady-state result for case ii. How long does it take for all the temperatures

to get within 1% of the steady-state result.

d) Redo part (a) or (b) for case ii but with 21 points rather than 11. Does this significantly

change your results?

Bonus Credit (5 points): Reformulate the finite difference 1D time-dependent forms for the

case where k is not constant. Redo case ii explicitly assuming that:

(T 300)

k (T ) 7 4

200

but assuming h and a remain constant. Compare your results to the previous results. 4. (40 points)

a) Analyze the steady heat transfer characteristics of the given fin design computationally

using a grid spacing of 1.5 cm. Determine the total fin heat transfer per unit width, the fin

effectiveness, and the fin efficiency. The base temperature is 90oC, the ambient temperature

is 5oC, the fin material has a thermal conductivity of kf = 40 W/m-K, and the convection

coefficient is 80 W/m2-K. You are welcome to use symmetry arguments where appropriate.

b) Compare the heat transfer per unit width to the following analytical solutions:

i) A triangular fin of the same base thickness (9 cm) and length

ii) A triangular fin of the same base thickness and profile area

ii) A rectangular fin of the same length and constant thickness of 6 cm

Which analytical approach most closely matches the computational result?

c) Plot the computationally determined centerline temperature of the fin in terms of Tf vs. x.

For comparison, plot on the same figure the following analytical temperature profiles:

i)

A rectangular fin of the same length and constant thickness of 9 cm

ii)

A rectangular fin of the same length and constant thickness of 6 cm

iii) As (i) and (ii) above, but with constant thickness 3 cm

iv) A compound fin solution ? solve the first fin segment as a fin of length 6 cm and a

convective tip. Use the tip temperature of this segment as the base temperature of the

next rectangular segment, and repeat through the third segment.

Which analytical approach yields the best comparison with the computed result? h 80 W/m2 K

T 5 o C

k f 40 W/m K

Tb 90 o C 9 cm 6 cm 6 cm 6 cm 3 cm 6 cm 4. Continued

Continuing with the fin problem, assume the fin is starting with a uniform temperature of Ti

= 5oC. Analyze the development of the temperature field within the fin over time using either

the implicit or explicit method. Plot the centerline results at intervals of DFot = 0.5 (i.e. Fot

= 0, 0.5, 1.0, 1.5, 2.0?), etc. up to Fot = 5. Show the full temperature field at Fo = 5. How

does the temperature field result at Fot = 5 compare to the steady-state result? Compute the

heat transfer per unit width through the fin at Fot = 1 and Fot = 5 and the fin effectiveness

and fin efficiency at these times. Bonus Credit (10 points): Redo both your steady (5 points) and unsteady (5 points) analysis with a

grid spacing of 0.75 cm. Recalculate your heat transfer per unit width, the fin effectiveness,

and the fin efficiency. Also plot your centerline temperature profile versus the results from

(c). How do higher resolution grids change your results? h 80 W/m2 K

T 5 o C o Tb 90 C Fot Ti 5 o C

k f 40 W/m K Lb 9 cm

6 cm f 7800 kg/m 3 3 cm c f 400 J/kg K 6 cm 6 cm 6 cm at

( Lb / 2) 2