Ch.E. 531 - Heat Transfer - Homework No. 8

due Tuesday, Mar. 6

24. Determine the units of the constants in the k-e model. Consider Figure 13.6. For flow of water in a smooth pipe at a Reynolds number of 10⁴, identify the actual distance (i.e. convert from y+ to y).

Do the same thing for air at Re = 10⁵.

25. A hot wire anenometer might use a platinum wire 0.5 mm in diameter. If it is to be used to measure air velocity at about 20 m/s, what is the expected heat transfer coefficient? How would the heat transfer coefficient vary with velocity?

26. Solve the two-equation model of turbulence given in class. By changing the initial condition and the Reynolds number, see what wierd behavior you can obtain. Extra credit will be given to the person with the most interesting (and artisitic) diagram.

27. Figure 13-7 shows the velocity profile for flow in a tube at two different Reynolds numbers. Use FIDAP to compute a turbulent flow using a k-e model and compare with this Figure. Your goal is to say whether FIDAP gives correct results or not. Use Example 19 as a starting point, but each person will use a different model (see Vol. 5 of the documentatio).

Standard k-e -Chen, Dai

extended k-e - Schumacher

RNG - Zheng

Wilcox low-Re - Sun

Be prepared to present your results and conclusions in class.

Ch.E. 531 - Heat Transfer - Homework No. 7

due Tuesday, Feb. 27

21-22. Solve the problem of convection in a horizontal layer of porous material filled with liquid. The momentum equation is replaced by Darcy's Law, which is given as the first equation in Problem 12.7. The layer is heated from below, and the normal velocity at the top and bottom surface is zero. There are no other velocity boundary conditions. Your solution should include: start with the equations, make them non-dimensional, define the Rayleigh number for this problem, prove exchange of stability (if it applies), either solve exactly or approximately (one term in velocity and temperature is sufficient) for the Rayleigh number, and determine the wave number at the critical Rayleigh number. The entire problem parallels the one done in class.

23. Consider a fluid layer of water (not in a porous media). The bottom surface is rigid and the top surface is free (exposed to air). Thus, surface tension and gravity both apply. Use the correlation given in class and insert the critical Rayleigh number, the critical Marangoni number, and derive an equation for the critical temperature difference when both mechanisms are important. Determine the depth of fluid layer that will make the surface tension effect by 99% of the effect; do the same for gravity. How would the surface differ in these two cases, assuming a hexagonal pattern resulted from both of them.

Ch.E. 531 - Heat Transfer - Homework No. 6

due Tuesday, Feb. 20

18. In Ex. 9.5-1, the Nusselt number at infinity is l₁²/2. The eigenvalue l₁² is defined as the

smallest eigenvalue of this problem:

One way to solve such eigenvalue problems is to use the Calculus of Variations. In the variational method (which is the basis of the original finite element method), the eigenvalue is

among all functions F(h) that satisfy the boundary conditions and are orthogonal to all the eigenfunctions corresponding to other eigenvalues.

Thus, for the first one we merely need to calculate

with some approximate function for F(h). Do so for F(h) = a (1 ­ h²). Why is this appropriate? If you were to write F(h) = S_i a_i f_i(h), what would you use for the second term?

19. Change your data set for the Graetz problem to compare the case of an inlet velocity of

versus u(y) = 1. The data set we had used u(y) = 1. To make it solve for the fully developed velocity profile, we change the command

BCNODE(UZC,CONSTANT=1.0,ENTITY="inflow")

to

BCNODE(UZC,ENTITY="inflow",POLYNOMIAL=1)

2.0 ­2.0 0 2 0

To calculate the overall heat transer for both cases, choose Calculation, Flux, and choose the entity of the "wall". Compare the overall flux for both cases. You can plot the flux along the wall by issuing the command (in the gui): flux(temperature,diffusive,plot,entity="wall").

20. Perform a one-term approximate analysis of the Rayleigh problem with rigid boundaries. The equations are Eq. (12.3-38,39). Set the time constant s to zero. To solve these equations, derive an expansion for temperature that is a polynomial and satisfies the temperature boundary conditions, T = 0 at z = 0 and L. Derive an expansion for velocity that is a polynomial and satisfies the velocity boundary conditions, v = Dv = 0 at z = 0 and L. (See problem 5 on the test.)

Substitute these expansions into the revised Eq. (12.3-38, 39). Find the conditions under which a non-trivial solution exists. (Hint: the result will be of the general form, but more complicated, as Eq. 12.3-5). Determine what the critical Rayleigh number is for instability, and the wave number at that critical condition.

Ch.E. 531 - Heat Transfer - Homework No. 5

due Tuesday, Feb. 13

14. Run the CFD program FIDAP with the example data set given in class Thursday. Prepare a plot of some feature of the solution. (Later we will solve the problem in which both the velocity and the temperature are changing at the inlet.)

15. Look at all the examples in the FIDAP set of examples. Choose one that is closely aligned with your research interests. Propose (first draft) a problem to be solved using modifications of that data set as a project problem.

16. Using values from Figure 9-5 for the constant wall temperature case, prepare a correlation for Nu versus z that is valid over all z. Plot Nu versus z on a linear plot (i.e. not a log-log plot). The correlation should be of the form

To save time, you can choose n based on a single point (carefully chosen) from Figure 9-5.

17. Estimate the heat loss over a three month period (Watts) through a window pane that is 0.8 m wide and 0.6 m high. Assume that the temperature of the outside air is always 30 K lower than the desired temperature inside. The glass that is available is 0.32 cm thick and has a thermal conductivity of 0.69 W/mK. Allow for a heat transfer coefficient of 10 W/m²K on both the outer pane and innermost pane, i.e. accounting for the heat transfer between the glass and outside air, and between the glass and inside air. Give the heat loss in watts for each of these options:

1. one pane of glass

2. two panes of glass, separated by 5 mm and filled with argon at one atmosphere

3. two panes of glass, separated by 5 mm and filled with dry air at one atmosphere

4. three panes of glass, with gaps of 5 mm between each pair, and filled with argon.

Ch.E. 531 - Heat Transfer - Homework No. 4

due Tuesday, Jan. 30

9. Eq. (10.2-10) is an expression derived for low Reynolds number using an asymptotic analysis. Compare the following:

(a) Eq. (10.2-10) using only the terms up to linear in Péclet number;

(b) Eq. (10.2-10) using all terms given there; and

(c) using the following expression:

Make the comparison by plotting Nu versus Pe on a log-log graph, using 0.1 < Pe < 10 for (a) and (b) and 0.1 < Pe < 1000 for (c). Make a second comparison in the same way, except plot Nu ­ 2 versus Pe.

10. Pick a liquid and find its properties so that you can calculate the Prandtl number at normal temperature and pressure. Which of the properties varies the most if the temperature is changed 10 °C? Pick a gas and do the same thing. If the gas were at very low pressure (vacuum), how would the properties (and Prandtl number) vary with temperature and pressure?

11. Look on the Web and report in class (one minute length) about something you found about the finite element method. It can be methodology or applications. Possible key words are "finite element method", "computational fluid dynamics", "heat transfer", "fluid flow".

12. Consider the related problem of unsteady diffusion in a slab geometry. Use your numerical code (modified slightly from problem 2) to solve a diffusion problem in three cases: one with a constant diffusivity, one in which the diffusivity depends exponentially on the concentration [exp(ac)] in an increasing manner, and one in which the diffusivity decreases with increasing concentration [1 - ac]. Your goal is to show qualitatively how the concentration profiles change when the diffusivity varies with concentration. Discuss your results.

13. Run pde toolbox as you did for Problem 6. After running the code, plot the contours and save the vectors u, p, e, and t. The variable u is the solution. The variables p, e, and t are the points (x and y coordinates), the edges, and the nodes in each element, respectively. Identify one element in your plot. List the appropriate u, p, e, and t matrices (i.e. the ones relevant to that triangle). Make a sketch of that element, giving the x and y coordinates of each corner, the node numbers, and the element number.

Ch.E. 531 - Heat Transfer - Homework No. 3

due Tuesday, Jan. 23

7. If you haven't finished Problem No. 3, do so.

8. Verify the following equations for Taylor-Aris dispersion if they weren't covered in class: 9.7-7, 11, 16, 17, sentence after 9.7-22, 9.7-24, 25, 26, 30, 31. Hints: for 22, show that with C₀ = m₀ the integral is zero; for 24, first write down the differential equation that satisfies, then show that you can find a solution that is only a function of eta; for 25, try .

Ch.E. 531 - Heat Transfer - W, 2001 - Homework No. 2

due Tuesday, Jan. 16

4. Problem 9.1

5. Verify the following equations for the Leveque problem if they weren't covered in class:

9.4-15, 16, 21, 24, 27, 28

6. Solve the same Problem No. 3, but using the pde toolbox in Matlab. Prepare a movie of the transient results.

Ch.E. 531 - Heat Transfer - W, 2001 - Homework No. 1

due Tuesday, Jan. 9

1. Write a one page summary of what the paper by Newman covers.

2. Reproduce Figures 2-3-4 and Figure 5 of the paper by Newman.

3. Use the ADI method to solve the same problem shown in Figure 2 of the handout describing heat transfer during cooking. The disk is 10 cm in diameter, 2 cm thick. Initially the meat is at

10 °C and the boiling oil is at 100 °C. Take the heat transfer coefficient as 1000 W/m² K, the thermal conductivity as 4 W/m K and rC_p = 1.4 x 107 J/m³ K. Integrate until t = 360 s. How well do your results compare with Figure 2?