M I D - E X A M

Special Problems in Nuclear Thermal-Hydraulics

Oct. 29  2008

PART 1 (Take-Home EXAM)

- Choose 3 Problems and solve. Handout by 9:00 AM on

Oct. 30 to Jong-Won Kim.

Don't excuse the due time !!!!!

1. Derive the nodal one-dimensional two-phase conservation equations for mass, energy and momentum.

Assume the upper boundary of the node is fixed, but the

lower one is moving[e.g. the boiling boundary, λ0(t)].

Note : Relate the node-average parameters to the

arithmetic average of the corresponding node inlet

and outlet parameters.

2. Consider a flat plate over which is flowing a liquid film with    no entrapped vapor.  Show that the appropriate normalized expression for the

liquid film flow rate per unit depth([eqn: w _{lf} ^{'}]) is given by:

[eqn: w _{lf} ^{+} ````=`` int _{0} ^{delta ^{+}} {u _{lf} ^{+} ``dy ^{+}}]

where, as usual,

[eqn: w _{lf} ^{+} ```` == ` {w' _{lf}} over {mu _{l} ``g _{c}}] ,  [eqn: u _{lf} ^{+} ```` == ` {u _{lf}} over {u ^{*}}] ,  [eqn: y _{} ^{+} ```` == ` {y rho _{l} `u ^{*}} over {mu _{l} ``g _{c}}] ,  [eqn: u ^{*} ```` == ` sqrt {{tau _{w} `g _{c}} over {rho _{l} ``}}]

Given the "universal velocity profile";

[eqn: u ^{+} ``=``y ^{+} ``````(`y ^{+} ``` LEQ ``5)]

[eqn: u ^{+} ``=`-3.05``+``5.0``ln`y ^{+} ``````(`5`` LEQ `y ^{+} ``` LEQ ``30)]

[eqn: u ^{+} ``=`5.5``+``2.5``ln`y ^{+} ``````(``y ^{+} ``` GEQ ``30)]

Calculate [eqn: w _{lf} ^{'}] for water at 100oF and atmospheric pressure, if the measured

normalized liquid film thickness is [eqn: delta ^{+`} ``=``39]

3. The van Driest's eddy viscosity profile is given as,

[eqn: epsilon ^{+} ``=`1+ {epsilon } over {nu } `=` {1} over {2} `+` {1} over {2} LEFT { 1.0`+`4 kappa ^{2} ``y ^{+2} `[1.0`-`exp(-y ^{+} /26)] ^{2} RIGHT } ^{1/2} `]

where [eqn: kappa ``=``0.41] and [eqn: Pr _{t} `=`1.0] ,

Show that the Stanton number for turbulent flow in a pipe in the limit as Pr goes

to infinity is

[eqn: St``=`0.113`(C _{f} `/`2) ^{1/2} `Pr ^{-3/4}]

Note that, for large Pr, the temperature profile near the wall is very steep. Thus,

you can assume a constant heat flux near the wall, and retain only the first term

in a Taylor series expansion of [eqn: epsilon ^{+}]

4. We want to predict the average departure diamter, Db , of the bubble and its

frequency, f , all following the general rule :

[eqn: rm D _{b} ^{m} `f ^{n} ``=``constant]

Consider the data presented on the next figure concerning the actual frequency

distribution of departing bubbles from a platinum wire, where F(Db)dDb is the

number of bubbles leaving the surface per nucleation site per unit time with

diameters between Db and dDb . Assume that the water is saturated at 14.7 psia.

Compare the experimental data with (a) Zuber's model, (b) Cole's model, (c)

Mikic's model for the average departure diameter and the average departure

frequency.

[eqn: <D _{b} >``=`` {SIGMA F(D)D _{b} `dD _{b}} over {<f>}# # <f>```````=``` SIGMA F(D)`dD _{b}]

PART 2 (Closed Book)

- 10:30 ~ 11:50

Describe the followings ;

1. interfacial jump condition

2. virtual mass

3. IAC and Sauter diameter

4. critical Weber number

5. strength and weakness of two fluid model

6. nature of turbulence

7 Process of derivation of steady turbulent conservation equation

8 the reason why u'v'can not be 0

9 (1) k-ε model and its weakness  (2) behaviour of k near the wall

10. LES model