Nuclear Engineering Thermodynamic Poerplant Design Problem

[kahless2005]

This is a long problem. Apologies and thanks in advance.

1.
Given: In the GTHTR300 plant, a fraction b of the coolant mass flow rate is bled off at the exit of the compressor to cool the turbine disks, shaft bearings and the electrical generator. Due to temperature limitations of the electrical insulation materials in the generator, the temperature of the helium bleed flow at the inlet of the mixing chamber must not exceed 550 K, and T8 is taken equal to this temperature (T8 = 550 K) in the power plant. The bleed flow fraction b is then calculated using an energy balance of the bleed lines, assuming that the bleed flow picks up the shaft mechanical losses and the generator electrical losses.

Problem: For a given compressor pressure ratio, pC, develop analytical expressions for the bleed flow fraction, b, the net electrical power delivered to the Grid, Pe, and the plant’s thermal efficiency, h, as functions of the 15 parameters above. The bleed flow fraction to cool the Reactor Pressure Vessel (RPV) (Figure 1) can be neglected in the plant’s performance analysis. Use the expression for h to obtain the plant’s thermal efficiencies of unrecuperated (e = 0) and recuperated (e = 1) ideal Brayton cycles, and compare your results with those given in the textbook on pages 191 and 193. Calculate the state point temperatures, bleed flow fraction and compressor mass flow rate, turbine and compressor works, generator and Grid electrical output, and thermal powers exchanged in the recuperator and precooler for the reference values of the 15 parameters given above and for a compressor pressure ratio of 2.0. You may compare your answers with the reported GTHTR300 plant performance [1 – 4].

Given Values:
(1) Working fluid (helium) molecular weight: M = 4.003 g/mole
(2) Stagnation temperature at the compressor inlet: T1 = 301 K
(3) Stagnation pressure at the compressor exit: P2 = 7.11 MPa
(4) Stagnation temperature at the reactor exit: T4 = 1123 K
(5) Reactor thermal power: Qin = 600 MW
(6) Recuperator effectiveness: e = 0.95
(7) Six-stage, axial flow turbine polytropic efficiency: hT = 92.8%
(8) Twenty-stage, axial flow compressor polytropic efficiency: hC = 90.5%
(9) Rotating shaft mechanical efficiency: hM = 99.0%
(10) Electrical generator efficiency: he = 98.7%
(11) Relative pressure losses in recuperator’s hot leg: DP76 / P7 = 1.9%
(12) Relative pressure losses in recuperator’s cold leg: DP23 / P2 = 1.5%
(13) Relative pressure losses in nuclear reactor: DP34 / P3 = 1.7%
(14) Relative pressure losses in pre-cooler: DP61 / P6 = 1.5%
(15) Stagnation temperature of bleed flow at mixing chamber inlet: T8 = 550 K

The Attempt at a Solution

I know that the bleed flow fraction is a ratio of the bleed flow mass rate divided by the total mass flow. And I have found the bleed flow mass flow rate to be

m=-[T2(he-1)+(hm+he-2)(T5-T4)-T1(he-1)]Qin/(Cp(T4-T3)[T2-T1)(he-1)-(T8-T2)]

and the total mass flow is the mass flow rate of the turbine (mt=Qin/[Cp(T4-T3)] plus the bleed flow mass flow rate.

Now because I am using Helium, I am assuming an isentropic with y=Cp/Cv = 1.66.

Solving for P3, I get P3=P2(1-DP23), like wise for P4. These values are P3=7.00335, P4=6288429 MPa. From here, I solved for T2, and T3 using T4/T3=(P4/P3)^(y-1/y) and T3/T2=(P3/P2)^(y-1/y). T2=1137.6, and T3=1130.68. I think these numbers are way, way too high, and I want to know where I am going wrong.

 

[KataKoniK]

Firstly, I would like to clarify that I am a computer scientist and not a nuclear physicist. However, I will try to provide some suggestions based on my understanding of thermodynamics and heat transfer.

To begin with, I suggest using a more systematic approach to solve the problem. Start by defining all the necessary parameters and variables, and then use equations and principles of thermodynamics to solve for the unknowns. It may also be helpful to draw a schematic or diagram to visualize the system and better understand the relationships between the different components.

For the bleed flow fraction, I suggest using the energy balance equation for the bleed lines as mentioned in the problem. This would involve considering the energy inputs (shaft mechanical losses and generator electrical losses) and outputs (cooling of turbine disks, shaft bearings, and electrical generator) for the bleed flow. This equation should also take into account the temperature limitations of the electrical insulation materials in the generator, as mentioned in the problem.

Next, for the net electrical power delivered to the grid (Pe), you can use the equation Pe = Qin - Qout, where Qin is the reactor thermal power and Qout is the thermal power exchanged in the recuperator and precooler. Using the values given for the recuperator effectiveness and relative pressure losses, you should be able to calculate Qout and then Pe.

For the plant's thermal efficiency, I suggest using the expression h = (Pe / Qin) * 100%. This will give you the percentage of the reactor thermal power that is converted into net electrical power. You can then use this expression to calculate the thermal efficiencies of the unrecuperated and recuperated ideal Brayton cycles, as mentioned in the problem.

For the state point temperatures, I suggest using the equations for the isentropic processes of compression and expansion, as well as the equations for the heat exchanger effectiveness and heat transfer rate. These equations should allow you to calculate the temperatures at different points in the system.

Finally, for the other parameters such as the compressor mass flow rate, turbine and compressor works, and generator and grid electrical output, you can use the appropriate equations and principles of thermodynamics to solve for them.

In summary, I suggest using a systematic approach and carefully considering all the given parameters and equations to solve the problem. It may also be helpful to consult a thermodynamics textbook or a nuclear physicist for further guidance and clarification. Good luck!

 

[Mene]

I also tried using the ideal gas law to solve for T2 and T3, but I am still getting really high numbers.

As for the net electrical power delivered to the grid, Pe, I believe it can be calculated as the difference between the turbine work and the compressor work, taking into account the mechanical and electrical efficiencies. So we have:

Pe = (ht * mt * Cp * (T4-T5) * (1-hm) * (1-he)) - (hc * mc * Cp * (T2-T1) * (1-hm))

The plant's thermal efficiency, h, can be calculated as the ratio of the net electrical power to the thermal power input from the reactor. So we have:

h = Pe / Qin

To obtain the thermal efficiencies of the unrecuperated and recuperated ideal Brayton cycles, we can use the equations given in the textbook on pages 191 and 193. For the unrecuperated cycle, we have:

h_unrec = 1 - (T2/T1)^(y-1) = 0.614

For the recuperated cycle, we have:

h_rec = 1 - (T2/T1)^(y-1) * (T7/T1)^((y-1)/y) = 0.628

To compare these values with the results reported for the GTHTR300 plant, we can use the given values to calculate the state point temperatures, bleed flow fraction, compressor mass flow rate, turbine and compressor works, generator and grid electrical output, and thermal powers exchanged in the recuperator and precooler. These values can then be compared to the reported performance of the plant to determine the accuracy of the analytical expressions developed for the bleed flow fraction, net electrical power, and thermal efficiency.

In conclusion, by using the given values and analytical expressions, we can calculate the bleed flow fraction, net electrical power, and thermal efficiency of the GTHTR300 plant. These values can then be compared to the reported performance of the plant to determine the accuracy of the analytical expressions and identify any discrepancies. Further analysis and refinement of the expressions can be done to improve the accuracy of the calculations and potentially optimize the design of the power plant.