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Unformatted text preview: NEEP 411: HW#3
Transient Fuel Rod Conduction
Due: Friday, October 7th, 2011 1. In Chapter 4 of ElWakil’s text, there is an empirical expression (433, page 96) for
decay heat, valid for times after shutdown exceeding 200 seconds: J— : 0.095t‘0‘26 t[s], z 2 200 Q0 We need an expression that extends back to t = 0 to use in upcoming simulations. A
semiempirical expression for decay heat might look like: 251 : Clexp(—/iqt) + Czexp(——Azt) + C3 expo—ﬂat) 0 S t [s] S 200 This represents the actual decay heat, in that it is a superposition of exponential terms,
but there are only three “groups” instead of the actual hundreds of contributions from
individual ﬁssion products. Suppose we let the halflives of these three groups be 10,
100 and 1000 seconds respectively. Find the weighting coefﬁcients C1, C2 and C3
such that: (a) the decay power is 0.07 x full power at t = 0 seconds; (b) the decay
power matches ElWakil’s Eqn (433) at t = 200 seconds; (0) the time rate of change
of the decay power matches the corresponding time rate of change of ElWakil’s
expression at t = 200 seconds. After you ﬁnd the coefﬁcients, plot the composite
decay heat over the interval from 0 to 500 seconds to verify the smoothness of your decay heat curve. 2. One of the inputs to our transient simulation is a heat sink time constant, 1w. This is the time for the heat sink (water) to undergo a signiﬁcant change in temperature. To
calculate this value, use the composite expression for decay heat that includes your results from Problem 1. Given the following approximate values for water properties
(/3 = 750 kg/m3, cp = 5200 J/kg°C), ﬁnd the time required to raise the water
temperature 50 °C. Assume there is 350 m3 of water in the primary system of a PWR
that produces a thermal power (before shutdown) of 3000 MWth. This corresponds to
the time required to raise the bulk temperature in the primary system to the saturation
temperature. (Hint: the time required exceeds 200 seconds, so you can integrate the
exponential expression from Problem 1 from 0 to 200 seconds and add it to whatever
is required from El—Wakil’s expression.) 3. These last two problems, 3 and 4, involve getting ready to run and then actually
running a transient simulation of a fuel rod with Matlab. The reason we have to
spend some time “getting ready to run” has to do with the speciﬁcs of our problem.
We are dealing with extremes in this problem: extremely small dimensions (ms) and extremely large temperatures (~1000 °C). For this reason, it is useful to ﬁrst cast the problem in dimensionless form. We want to represent the spatial domain in terms
of dimensionless variable x = r/R, so that the spatial domain is O S x S 1. We want to normalize the time to our fuel time constant, 77 = t/zy,
1 R2 1 R2
7f = 2 “‘— z ——*
(11R) 05 4 a We should also normalize the temperature to a peak value from our steadystate
solution: HI — 2
f=——T, gar#9 (“R
T 4k max Beginning from the actual heat conduction equation and its initial and boundary
conditions, 16(r8T)+q (rd)=16T
7'61” 0r k a ﬁt
qm(0_) 2 2 1 —
IC:Tr,t=0 =T + R — , TS=Tw+—— " 0
( ) 3 4k ( r) U 61( )
BC: g—(r=0,t)=0 ; —~k(—3]:(r=R,t)=U[T(r=R,t)—Tw]
6r 6r rewrite the governing equation, initial and boundary conditions in dimensionless form, T (r,t) ) T (x, 77) . You can verify that you have the right form by consulting
the Matlab script posted on the eCOW2 web page, trans_fuel_rod.m. Keep in mind
that Matlab is expecting the problem to be written in a particular form, as indicated in
the Notes about the pdepe solver also posted on the eCOW2 web page. . In the Matlab script, trans_fuel_rod.m, provided on the eCOW2 web page, modify the
expression for decay heat to account for your results in Problem 1. Also, the Bi
number and water temperature are both constants in the trans_ﬁ1elwrod.m script.
Suppose now we account for a pump coastdown time by representing Bi(t) as a function of time:
Bi(t) = Bi0 exp(~t /' 1p) + Bit,o [l eXp(—t/rp )] The Bi won’t actually go to zero as we lose the pumps, in part because there is still
some heat removal even in stagnant water and in part because natural circulation currents will develop in the system (more on that later). Let Bio, = 0.1 x Bio and
choose 100 seconds for IP. Finally, represent the heat sink as: T w(t) = T w + AT(t/rw), where TWO is the initial
water temperature of 300 °C, AT is 50 °C, and Tw is the time you evaluated in
Problem 2. Modify the Matlab script to incorporate these changes (decay heat, timedependent Bi
and water temperature) and run the simulation out to thousands of units of 77. How
long does it take for the fuel centerline temperature to return to its peak value before the transient started? ...
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