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GENG4405 - Assignment 2
Part 1 Flash geothermal power plants work
in the following manner: • Liquid water exists in a high temperature reservoir, underground. The water is a saturated liquid or a subcooled liquid close to its saturation temperate and is under ~hydrostatic pressure. • By drilling into the reservoir and pumping, that water can be brought to the surface. Some pressure drop and consequently phase separation (i.e. flashing) is necessary. Increased pressure drop (that is to say, lower wellhead pressure) leads to an increase in flowrate until the well becomes choked at which point further pressure drop will not increase flowrate. • The water moves through insulated pipes along the surface from the wellhead to the powerhouse where it goes into a “separator” in which the saturated vapor is separated from the saturated liquid. • The steam from the separator goes into the turbine to generate power and then is condensed, usually in a water-cooled condenser. • The liquid from the separator can either be disposed of by reinjection to the reservoir through an injection well or flashed again at a lower pressure in a second separator, with the steam being fed to a second, low pressure turbine. Australia has the wrong geology for such power plants but they are common in New Zealand, Indonesia and the Philippines and there is significant potential on New Guinea. Part 1 of the Assignment: How big is the reservoir? A layer of highly permeable rock has been identified. Primarily geophysical measurements have led to a network of points, spaced at 500 meter intervals, at which the depth to the top and bottom of the reservoir are approximately known. The data has been passed to your team (as an Excel spread sheet) with the expectation that you’ll be able to estimate the volume of the reservoir. The data consists of the elevation above sea level of the surface of the ground at the point where each measurement was made, the depth below the surface at which the top of the reservoir was detected and the depth below the surface at which the bottom of the reservoir was detected. The measurements were made by seismic reflectance and it’s known that this form of measurement has trouble distinguishing between layers that aren’t at least 10 meters apart. A geologist has also looked at the data and she notes that the permeable layer appears to have filled in an ancient valley; the base rock has much lower permeability than the reservoir. After deposition the permeable layer was covered by another impermeable layer. The reservoir runs approximately along a north-south axis. Faulting has (conveniently) truncated the reservoir just south of the first row of data and just north of the ninth row. Less conveniently it appears that the ancient valley branched at the northern end and part of the lesser branch falls outside the grid of data. The geologist has also drawn her interpretation of the shape of the reservoir (fig. 1). Fig. 1 – Geologist’s interpretation of the shape of the reservoir Assignment questions: a. Report the volume as calculated using the trapezoid rule in your submission (5 marks) b. Report the volume as calculated using the Simpson’s rule in your submission (5 marks) For the volume integration, first use the method (either trapezoid or Simpson’s) on the depths in each row (1-9), to find the cross-sectional area for the row. Then use the same method on the cross-sectional areas of each row to find the volume of the reservoir. Explain in the report any assumptions your team has made. This is a case where there is a degree of uncertainty. Some of reservoir appears to fall outside the data grid and some of the data falls within the uncertainty limits of the measurement method (that is the difference in depths is less than 10 meters). Use your best judgement. Because of the degree of uncertainty, there is no canonically correct answer however any assumptions you make must be stated and justified. GENG4405 - Assignment 2, Part 2 Part 2 of the Assignment: What is the fluid volume? The geothermal fluid from the reservoir reaches the surface as a two-phase fluid because of the pressure drop. The vapor fraction will be separated off and run through a turbine and the liquid fraction will be reinjected to the reservoir. For the purposes of sizing pipes and equipment it is necessary to know the specific volume (on a mass basis) of the liquid and vapor phases. Your team has been tasked with developing code to calculate the volumes based on the pressure to be used in the separator and turbine inlet. While the final pressure has yet to be determined, through an optimization procedure, you can demonstrate that your code works by using 8 bar absolute pressure. The geothermal fluid may be approximated as pure water. Use the Soave-Redlich-Kwong equation of state1 to model the properties of water. The equation is cubic in volume and has 3 roots; however negative and complex roots have no physical meaning. The smallest, real root (if positive) is the molar specific volume of the liquid phase, the largest real root is the molar specific volume of the vapor phase. If there is a middle, real root, it has no physical meaning and can be ignored. (, ) = − − ∙ ( + ) 1. in which = 0.42748 2 2 = 0.08664 = [1 + (1 − 1 2)] 2 = = 0.37464 + 1.54226 − 0.269922 General properties of water = 647.1 K = 22.0664 MPa = 0.344 = 18.015 g/mol Ideal Gas Constant = 8.314 J mol K The following is for your information only and is not part of the assignment question. For saturated systems the temperature and the pressure are dependent on one another, however when trying to use the equation of state to calculate saturation properties there is no way of knowing ahead of time whether the temperature and pressure match the saturation conditions or not. In the case of the assignment the pressure is fixed (8 bar) but the temperature is a guess. To check that guess it is necessary to evaluate whether the calculated liquid phase volume and vapor phase volume satisfy the condition for equilibrium. The way to check for equilibrium is to see if the fugacity coefficient of the liquid phase is equal to the fugacity coefficient of the vapor phase and if they are not equal, adjust the temperature until they are equal to within an acceptable tolerance (secant method)2. − = 2. The fugacity coefficients can be calculated from () = − − () + ∫ ( − ) ∞ 3. Where Z is the compressibility factor = The fugacity coefficient expression is solved twice, once with = the liquid phase molar specific volume and a second time with = the vapor phase molar specific volume (both of which come from finding the roots of the equation of state). Assignment questions: a. Plot the function (,) − 800,000 Pa = 0 (equation 1) on a P-v diagram and show the approximate location of the roots for an initial guessed temperature of 175 ℃. Use a logarithmic scale for the volume axis with a range between 10−5 and 10−2 and an appropriate range for the pressure axis. Label the axes of the figure and include appropriate units. Put this plot in your submission (7.5 marks) b. Using your graph as a guide, find the roots corresponding to the liquid and vapor phase volumes at 175 ℃ and 165 ℃ using either bisection or the secant method. Report your calculated specific volumes (on a mass basis) for liquid and vapor at 175 ℃ and 165 ℃ in your submission (7.5 marks) Explain in the report any assumptions your team has made. References: 1. Soave, G.; Equilibrium Constants from a Modified Redlich-Kwong Equation of State, Chem. Eng. Sci., Vol. 27, No 6 1972 2. Segura, H., Wisniak, j.; Calculation of Pure Saturation Properties Using Cubic Equations of State, Computers Chem. Engng., Vol. 21, No. 12, 1997 GENG4405 - Assignment 2, Part 3 Part 3 of the Assignment: What is optimal pressure? The geothermal fluid enters the separator as a two-phase fluid. After separation, the saturated liquid is reinjected to the reservoir and the saturated vapor is run through a turbine and then condensed and reinjected. The power output of a turbine is dependent on the enthalpy change of the steam running through the turbine and the mass flowrate of the steam ̇ = ̇ ∗ (ℎ − ℎ) Since in the case of our geothermal reservoir the steam comes in as a saturated vapor, its enthalpy is a function of the saturation pressure. The enthalpy leaving the turbine is a function of the condenser pressure (which is in turn a function of the cooling water temperature and temperature difference across the condenser). The mass flowrate is also connected to the pressure. The reservoir itself is at near hydrostatic pressure (and the pressure in the reservoir will likely decrease over time if the water is taken out faster than it can recharge). Hydrostatic pressure would imply that under no-flow the water level in the well would just reach the surface, where it would be at atmospheric pressure. To get water to flow, a pump is placed down in the well to boost the pressure in the region of the well above the pump. By controlling the pressure in the separator, the overall pressure drop between the reservoir and the turbine inlet can be adjusted. This controls the mass flowrate of water entering the well. Therefore, the power output of the turbine can be optimized by adjusting the pressure of the steam coming in (which will affect both the flowrate and the enthalpy change). A 1700-meter-deep well has been drilled into the reservoir. You may assume hydrostatic pressure. The water in the reservoir may be assumed to be saturated liquid. For our purposes the process of bringing the geothermal fluid to the surface and introducing it to the separator may be approximated as isenthalpic (another assumption). You may assume the pressure in the condenser is 10 kPa. The turbine has an isentropic efficiency of 0.85. Your team has been given the task of finding the pressure which will maximize the power output of the turbine. A test has been conducted linking the flowrate in the well to the pressure of the separator. That test data is included in this document. Steam tables will be provided on LMS. Separator Pressure in MPa Mass Flowrate in kg/s 0.2 138 0.3 138 0.4 135 0.5 129 0.6 122 0.7 113 0.8 93 1.0 64 1.2 36 1.4 0 Suggestion: Create a polynomial function describing flowrate as a function of pressure (being sure not to overfit the data, which may have some noise). You might also consider the same approach for the steam table data (paying close attention the range of data you actually need). This does not have to be done in MATLAB and is not a part of the assignment question. Assignment Question: Write MATLAB code which calculates: i. The total mass flowrate as a function of pressure (based on the data provided) ii. The quality of the steam in the separator (as a function of separator pressure) iii. The mass flowrate of steam into the turbine (the steam quality in the separator multiplied by the total mass flowrate) iv. The specific enthalpy of the steam entering the turbine (which is the enthalpy of the saturated vapor at the separator pressure) v. The specific entropy of the steam entering the turbine (which is the entropy of the saturated vapor at the separator pressure) vi. The specific enthalpy of the 2-phase water in the condenser if the turbine was isentropic (determine the quality at 10 kPa that has the same entropy as the saturated vapor at the turbine inlet, then calculate the enthalpy at 10 kPa based on that quality) vii. The difference between the specific enthalpy of the saturated steam entering the turbine and the enthalpy calculated in the previous step for the condenser viii. Multiply the mass flowrate of steam entering the turbine by the enthalpy change and by the turbine isentropic efficiency to get the power output With that code written, use one of the methods for optimization to vary the separator pressure in order to maximize the power output. a. Report the maximum power output of the turbine and the separator pressure needed to obtain this maximum value in your submission (25 marks) Explain in the report any assumptions your team has made. 1. Geothermal fluid, approximated as pure water, enters the well from the reservoir as (approximately) a saturated liquid at hydrostatic pressure. 2. The water moves up the well, through the surface pipework and into the separator. This is approximated as an isenthalpic expansion to the pressure of the separator. 3. The fluid splits into two phases and the saturated vapor is sent to the turbine. 4. The saturated vapor passes through the turbine and exits at the condenser pressure. Had it expanded isentropically, it would have reached point 4s, instead it reaches point 4. The enthalpy change between points 3 and 4 is 85% (the isentropic efficiency) of the enthalpy change between points 3 and 4s. If you need to refresh your knowledge of thermodynamics, a discussion of enthalpy and quality can be found in any standard text. For instance, Engineering and Chemical Thermodynamics by M.D. Koretsky. Alternatively, a copy of the ENSC2002 course reader (based on Koretsky) would also work. GENG4405 - Assignment 2, Part 4 Cooling water is supplied to the condenser of the powerplant by a positive displacement pump. During installation and significant maintenance operations the condenser can be isolated from the pump (cooling water inlet) and discharge line (cooling water outlet) via isolating valves. To help fill the condenser with water during commissioning and to allow for the removal of gases carried in with the cooling water, the high point in the condenser is connect, via a steel pipe, to a brass bleed valve on the roof of the turbine hall. The bleed valve is normally kept closed. During a maintenance operation a miscommunication leads to the isolating valve on the discharge line being closed before the pump was shut off. The resultant increase in pressure inside the condenser leads to the threads of the brass bleed valve shearing off, launching the valve straight up into the air at high speed. At the same time this maintenance is being carried out, a small plane is circling slowly over the powerplant and making observations at a height of 120 meters. If the valve hits the plane on the way up, what will be the energy of impact (i.e. 1 2 2 for the valve)? If instead the valve hits the plane as the valve falls back down, what will be the energy of impact? You do not need to consider the velocity of the plane. The elevation of the roof of the turbine hall (from which the valve was launched) is 6 meters. The initial velocity of the valve is 220 m/s. The mass of the valve is 550 g. The drag coefficient () of the valve is 0.29. The projected area of the valve () is 5.0 cm2 and the density of air () at the elevation of the turbine hall is 1.2 kg/m3. = 2 2 Assignment Question: • Report the energy of impact (10 marks) Explain in the report any assumptions your team has made. You may use the solvers built into MATLAB to do the stepping (ode23 or ode45 for instance) however other tools not addressed in the second half of GENG4405 are not permitted (for instance Simulink).
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