NE 418 Exam # Spring 13 1 (4% The piping system shown below is equipped with pressure and temperature measuring sensors. Temperature is measured by thermocouples and pressure is measured by strain gauges. T 1 T T 3 P 3 P 1 P D 1 D D 1 L 1 L L 1 a If the pressure measured by P is half of that measured by P 1, derive the equation that relates the velocity in the second section (v z to the inlet velocity (v z1. Do not neglect the gravitational effect (known as the potential energy per unit volume. Bernoulli s equation: 1 1 P + v + gh z = P + v + gh z = constant 1st section nd section 1 1 P + v + gh = P + v + gh 1 z1 1 z 1 1 v = z ( P P 1 + g( h h 1 + vz1 1/ P from which v = g z + ( h h 1 + vz1 b What is the quantity (Q of the liquid flowing into thend section? Q= va
1/ P π D nd section = ( = z = ( 1 nd section + + z1 Q va v A g h h v 4 c Denoting the viscosity of the fluid as μ fluid, write down the equation of the Reynold s number for this fluid as the fluid flows through the nd section v D D P ( z R = = g h h v e 1 z1 μ μ + + fluid fluid d As the pressure sensors are strain gauges, and as the pressure sensor P measured half of that measured by P 1, obtain the value of the resistance of gauge (R g with respect to that of gauge 1 (R g1. The batteries supplying the input voltage for the sensors are identical. Resistors in the bridges (in kω are R 1 =, R =6, R 3 =5, R 4 =8, R5=4, R 6 =8kΩ. 1/ 1 R 1 R g1 R 4 R g Battery Battery R R 3 R 5 R 6 P 1 sensor P sensor R1=, R=6, R3 =5, R4=8, R5=4, R6=8 Sensor 1 1 R3 R = R + R R + R Sensor 3 g1 1 R6 R5 = R + R R + R 6 g 4 5 input input
P 1 = = = P 1 1 3 g1 1 g R6 R5 R + R R + R 6 g 4 5 R3 R R + R R + R 3 g1 1 R6 R5 8 4 8 1 1 R6 + Rg R4 + R5 8+ Rg 8+ 4 8+ Rg 3 16 5 = = = = R3 R 5 6 5 8+ Rg 3 5+ Rg1 3 R + R R + R 5+ R + 6 5+ R 3 g1 g1 g1 16 5 = 4 + Rg = 8 + R g1 Rg = 4 + R 8 + R 5 + R g1 e A C-type thermocouple, which has a measuring range up to 5 o C measures the temperature in the 3 rd section. The relation between the temperature and the measured voltage across the thermocouple leads has the form: T = 7 + 43 +. 5. Determine the Seebeck coefficient of this thermocouple when the measured voltage is 1m. The Seebeck voltage is given by 1 α1 T 1 ( 7 43 1 5 1 1 = α + +. 1 α1 = = 57 3 o 19.74 1 m / Δ = Δ, hence = α1 T = α1 ( 7 + 43 +. 5 C (4% The one delayed group form of the reactor kinetics equations is dn ( β = n+ λc dc β n = λc a For < β show that shortly after a reactivity insertion, the reactor power increases according to the Prompt Jump Approximation dn λ n = β Shortly after the reactivity insertion, the solution for the neutron number density can by approximated by the solution of
i ( β = n+ λc ii dc β = n λc Rewrite i as ( β n =λ C and take the time derivative ( β dn dc =λ from ii dc β n = λc βn ( β = + n n = ( dn dc n β = λ = λ dn n = λ ( β b If the power increase follows a stable period of 81.5 seconds, determine the reactivity insertion From the solution to part a, the period is ( β τ = λ Solving for the reactivity β 1 = τ λ + λ 1 For the given data
1 1 β 1.75 1 = τ 81.5.1 λ + λ = + =.8.8 c Assuming the Prompt Jump Approximation is valid, and = α Q ( nn Show that a negative power coefficient is required for stability. Short Solution For dn n n = λ = λ ( β ( β and = α ( n n < β dn α ( n n n = λ β α ( n n For any perturbation in n such that n n = δ n > dn α n δ = λ n β α δ n For α ( + and α δn < β Q α δn λ > β α δ n which implies dn Q > and power increases without bound, such that the system is unstable. For α ( α δn λ < β α δ n which implies
dn < such that the system is unstable. Longer Solution Let n Fn (, = λ =λn( β ( β 1 Linearize about fixed reference values n, Fn (, Fn (, + ( n n + ( n n, n, n λ = β n, λn = 1 β β n, If the linearization is about an initially critical, steady state solution, then = and Fn (, = n n, n, = λn = β such that λn Fn (, β and dn n λ = where = α ( n n β β < For any perturbation in n such that n n = δ n > dn = αq λn δn β the right hand side of the equation is (- only for α <.
3 (% Short Answers Problem Data Prompt Neutron Lifetime 1-4 seconds Delayed Neutron Fraction.75 Delayed Neutron Decay Constant.8 seconds -1 a Why do thermocouples vary in their range of temperature measurements? ANSWER: Because they are made of different metals and hence they have different Seebeck coefficients b Why may thermocouples need to be connected to Op Amps? ANSWER: Because the output voltage can be small and must be amplified c What is a fiber optic pressure sensor? ANSWER: It is a sensor that a diaphragm that deflects under external applied force causing light intensity to change, thus measures the force/pressure d Where should we measure the flow in a PWR nuclear reactor? ANSWER: In all piping circulating coolants, steam, at the inlets and exits of the pumps e What is a variable area flowmeter? ANSWER: It is a meter that measures the change in the height of a float, which is directly proportional d = Q Δ t/ A to the flow rate ( f What is meant by prompt critical and what challenges does it present for reactor control? ANSWER: The reactor is critical on prompt neutrons alone, with a period on the order of fractions of a second. This implies any control system would have to initial control actions in this same time frame. g What is the basis of the F peak clad temperature limit? ANSWER: This is the threshold temperature for the exothermic metal-water reaction between Zirconium cladding and water h How is a negative fuel temperature coefficient ensured in UO fuels? ANSWER: The negative fuel temperature coefficient is the result of increased parasitic absorption of neutrons in U-38 at elevated temperatures. By utilizing low enrichment (low U-35 content relative to U-38 U in UO fuels, a negative fuel temperature coefficient is ensured.