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hw03_11 - NEEP 411 HW#3 Transient Fuel Rod Conduction Due...

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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 El-Wakil’s text, there is an empirical expression (4-33, 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 semi-empirical 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 half-lives 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 El-Wakil’s Eqn (4-33) at t = 200 seconds; (0) the time rate of change of the decay power matches the corresponding time rate of change of El-Wakil’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 steady-state 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, time-dependent 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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