Modeling and Analysis of Dynamic Systems

Size: px

Start display at page:

Download "Modeling and Analysis of Dynamic Systems"

Gertrude Garrett
5 years ago
Views:

1 Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 66

2 Outline 1 Lecture 9 - A: Chemical Systems Example: Continuously Stirred Tank Reactor Simulation Simulation Objectives Simulation Setup Simulation Results G. Ducard c 2 / 66

3 - Simulation Outline Example: Continuously Stirred Tank Reactor 1 Lecture 9 - A: Chemical Systems Example: Continuously Stirred Tank Reactor Simulation Simulation Objectives Simulation Setup Simulation Results G. Ducard c 3 / 66

4 - Simulation Chemical Systems Example: Continuously Stirred Tank Reactor Definition A chemical reaction is a transformation during which reactants are transformed into products. Remark: during a chemical reaction, molecules may appear/disappear to be transformed in other molecules, but in all cases the atoms are conserved, and thus the total mass is preserved. Chemical reactions typically involve two reactants αa+βb γc +δd the integer coefficients {α, β, γ, δ} describe the stoichiometry of the reaction and the double arrow indicates that the reaction can evolve in both directions. G. Ducard c 4 / 66

5 - Simulation Chemical Systems: examples Example: Continuously Stirred Tank Reactor Combination sodium + chlorine sodium chloride 2 Na(s) + Cl 2 (g) 2 NaCl(s) Combustion burning of propane C 3 H 8 (g) + 5 O 2 (g) 3 CO 2 (g) + 4 H 2 O(l) burning of coal (carbon) gives carbon dioxide C(s)+O 2 (g) CO 2 (g) Other types of reactions include: Reduction Oxidation Precipitation reactions etc. G. Ducard c 5 / 66

6 - Simulation Chemical Systems Example: Continuously Stirred Tank Reactor Chemical reactions typically involve two reactants αa+βb γc +δd Chemical reactions are best described on a molecular basis n A, n B, n C, n D with numbers of mol (quantity of element) n A = 1 mol is molecules of that species (Avogadro constant), Molar mass M A is the mass of 1 mol of A. m A = n A M A. Concentration [A] is the number of molecules n A in a given volume V: [A] = n A V G. Ducard c 6 / 66

7 - Simulation Example: Continuously Stirred Tank Reactor Chemical reaction kinetics We define the reaction advancement ξ, as the variation of the reactants quantity: dn A α = dn B β = dn C γ = dn D δ = dξ The reaction rate or speed of the reaction, or formation speed is thus v f = dξ dt v f = dξ dt = 1 dn A α dt = 1 β dn B dt = 1 γ dn C dt = 1 δ dn D dt If you divide by the volume V of the mixed reacting elements, the volumic speed of the reaction is v = 1 V dξ dt = 1 d[a] = 1 d[b] = 1 d[c] = 1 d[d] α dt β dt γ dt γ dt where [x] denotes the concentration of element x in [mol/m 3 ] G. Ducard c 7 / 66

8 - Simulation Example: Continuously Stirred Tank Reactor Chemical reaction kinetics v = 1 V dξ dt = 1 d[a] = 1 d[b] = 1 d[c] = 1 d[d] α dt β dt γ dt δ dt The reaction rate is also defined as v = r[a] p [B] q where p and q are called partial reaction orders. Remark: the coefficients p and q are often not equal to the stoichiometric coefficients, must be determined experimentally. G. Ducard c 8 / 66

9 - Simulation Case of First-order Reaction Example: Continuously Stirred Tank Reactor Consider the reaction A+B C Assume the reaction is first order in both reactants, then the reaction rate is: v = r [A][B] since the reaction rate is also it yields v = d[a] dt d[a] dt = r [A][B] G. Ducard c 9 / 66

10 - Simulation Case of equilibrium or opposed reactions Example: Continuously Stirred Tank Reactor The forward reaction rate αa+βb γc +δd (causing element A to disappear ) is also defined as: v = r [A] α [B] β. and thus the concentration change rate of reactant A is obtained as : 1 d [A] = r [A] α [B] β α dt d [A] = α r [A] α [B] β dt Remark: in this formulation, the probability that the reaction takes place is proportional to the probability that the necessary number of molecules A and B are in contact (concentration). G. Ducard c 10 / 66

11 - Simulation Case of equilibrium or opposed reactions Example: Continuously Stirred Tank Reactor The backward reaction rate (causing element A to appear ) is also defined as: v + = r + [C] γ [D] δ and thus the concentration change rate of reactant A is obtained as : 1 d + [A] = r + [C] γ [D] δ α dt d + [A] = α r + [C] γ [D] δ dt Total rate of formation of the species A (in mol/(s m 3 )) d [A] = α ( r + [C] γ [D] δ r [A] α [B] β) dt G. Ducard c 11 / 66

12 - Simulation Example: Continuously Stirred Tank Reactor The value of the reaction constants r +, and r, depend on: the pressure, and most importantly on the temperature. An Arrhenius model is used r + = k + (ϑ,p,...) e E+ /(Rϑ) R = J/mol K: universal gas constant. The constant k + is referred to as the the pre-exponential factor, E + is the activation energy. The Boltzmann term: exp{ E + /(Rϑ)} indicates the fraction of all collisions that have sufficient energy to start the reaction. G. Ducard c 12 / 66

13 - Simulation Example: Continuously Stirred Tank Reactor r + = k + (ϑ,p,...) e E+ /(Rϑ) r/k Rϑ/E Figure: Arrhenius function. Remark: it reminds of a probability function; see influence of temperature G. Ducard c 13 / 66

14 - Simulation Example: Continuously Stirred Tank Reactor Similarly, the reaction kinetic for the backward reaction r = k (ϑ,p,...)e E /(Rϑ) The four parameters: {k +,k,e +,E } must be determined experimentally. G. Ducard c 14 / 66

15 - Simulation Outline Example: Continuously Stirred Tank Reactor 1 Lecture 9 - A: Chemical Systems Example: Continuously Stirred Tank Reactor Simulation Simulation Objectives Simulation Setup Simulation Results G. Ducard c 15 / 66

16 - Simulation Example: Continuously Stirred Tank Reactor Example: continuously stirred tank reactor (CSTR) Assumptions A+B C 1 The concentration [B] can be assumed to remain constant (amount of B in inflow and in tank always more than necessary for the reaction.) 2 The dissociation A+B C is negligible. 3 The mass m and the density ρ of the fluid in the CSTR are constant. 4 The CSTR is perfectly insulated, the only heat transfer occurs through the controllable heat exchanger. G. Ducard c 16 / 66

17 - Simulation Example: Continuously Stirred Tank Reactor Example: continuously stirred tank reactor (CSTR) [A i (t)],ϑ i (t), m i Q (t) m ρ c ϑ(t) [A(t)],[C(t)] LC [C(t)],ϑ(t), m o Figure: Chemical reactor. G. Ducard c 17 / 66

18 - Simulation Modeling of the Chemical Reactor Example: Continuously Stirred Tank Reactor Step 1: Define Inputs and Outputs Control input = Q(t): rate of heat transferred by the heat exchanger Outputs = concentration of C and temperature ϑ(t) Disturbances = [A i (t)] and ϑ i (t) Notice that m i = m o = m (constant level control = constant mass in rector) and V i = V o = V= m /ρ G. Ducard c 18 / 66

19 - Simulation Modeling of the Chemical Reactor Example: Continuously Stirred Tank Reactor Step 2: Energy reservoirs Using the assumptions mentioned, three reservoirs must be modeled: n A : the amount of species A in the CSTR, level variable [A]; n C : the amount of species C in the CSTR, level variable [C]; U: the internal energy in the CSTR, level variable ϑ. G. Ducard c 19 / 66

20 - Simulation Modeling of the Chemical Reactor Example: Continuously Stirred Tank Reactor Step 3: Conservation laws For species A, the conservation laws yield d dt n A(t) = V [A i (t)] V [A(t)] V k [B] e E/(Rϑ) [A(t)] for species C d dt n C(t) = V [C(t)]+V k [B] e E/(Rϑ) [A(t)] and for the CSTR energy d dt U(ϑ(t),n A(t),n B (t),n C (t)]) = Hi(ϑ i (t)) H(ϑ(t))+ Q(t) Notice that the concentration [B] is (assumed to be) a constant. G. Ducard c 20 / 66

21 - Simulation Modeling of the Chemical Reactor Example: Continuously Stirred Tank Reactor Step 4: Express of internal energy as a function of temperature and composition du(ϑ,n A,n B,n C ) = U ϑ dϑ + U n A dn A + U n B dn B + U n C dn C = ρvc v dϑ +H A dn A +H B dn B +H C dn C where H A, H B, and H C are the enthalpies of formation for the corresponding species. Remark: Neither C v nor H A, H B, and H C are assumed to depend on the temperature ϑ. G. Ducard c 21 / 66

22 - Simulation Example: Continuously Stirred Tank Reactor du = ρvc v dϑ +H A dn A +H B dn B +H C dn C du dϑ dn = ρvc v dt +H A dn A dt +H B dn B dt +H C C dt dϑ 1 dn = ρc v dt +H A 1 dn A V dt +H B 1 B V dt +H C V dt 1 du V dt 1 du V dt = ρc v dϑ dt dϑ 1 dt = ρ C v [ 1 V du dt with internal energy variation du(t) dt +H A d[a] dt H A d[a] dt +H B d[b] dt H B d[b] dt +H C d[c] dt dn C dt H C d[c] dt ] = Hi(ϑ i (t)) H(ϑ(t))+ Q(t) = ρ VC p (ϑ i (t) ϑ(t))+ Q(t) Variations of elements due to chemical reactions: d[a] dt = d[b] dt = d[c] dt = r [A] G. Ducard c 22 / 66

23 - Simulation Example: Continuously Stirred Tank Reactor ( dϑ(t) 1 dt = 1 du ρ C v V dt H A d[a] d[b] dt H B ( 1 1 = ρ C v V ρ Consider: C p C v for liquids V dϑ(t) dt V VC p (ϑ i (t) ϑ(t))+ 1 V + 1 ρ C v (H A +H B H C )r [A](t) = ϑ i (t) ϑ(t)+ τ dϑ(t) dt = ϑ i (t) ϑ(t)+ 1 ρc v Q(t) + V V ρ C v d[c] dt H C dt Q(t) (H A +H B H C ) r [A](t) ρ C v V Q(t) + τ H 0 C vρ ke E/(Rϑ) [A(t)] V where the residence time τ, the overall reaction rate k and the reaction enthalpy H 0 are defined by τ := V/ V, k := k [B], H 0 = H A +H B H C ) ) G. Ducard c 23 / 66

24 - Simulation Example: Continuously Stirred Tank Reactor Step 5: Inserting the last equation in the conservation laws yields: τ d dt [A(t)] = [A i(t)] ( 1+τ k e E/(Rϑ)) [A(t)] τ d dt [C(t)] = [C(t)]+τ k e E/(Rϑ) [A(t)] τ d dt ϑ(t) = ϑ i(t) ϑ(t)+ 1 Q(t) k ρc v +τ H 0 C V vρ e E/(Rϑ) [A(t)] where the residence time τ, the overall reaction rate k and the reaction enthalpy H 0 are defined by τ := V/ V, k := k [B], H 0 = H A +H B H C Remark: H x > 0 if energy is needed to form species x. G. Ducard c 24 / 66

25 - Simulation Example: Continuously Stirred Tank Reactor Static behavior of the CSTR: i.e., Q = 0, rate of heat transferred by the heat exchanger, ϑ i = constant, and [A i ] = constant. Heat removed by the in- and out-flowing mass flow Hflow(ϑ)+ Q chem (ϑ) = 0 where Hflow(ϑ) = m C p (ϑ i ϑ) Vke E/(Rϑ) Q chem (ϑ) = H 0 1+τke E/(Rϑ)[A i] G. Ducard c 25 / 66

26 - Simulation Example: Continuously Stirred Tank Reactor P 3 Hflow Q chem P 2 P 1 Rϑ/E Figure: Steady-state points as a function of temperature. G. Ducard c 26 / 66

27 - Simulation Outline 1 Lecture 9 - A: Chemical Systems Example: Continuously Stirred Tank Reactor Simulation Simulation Objectives Simulation Setup Simulation Results G. Ducard c 27 / 66

28 - Simulation G. Ducard c 28 / 66

29 - Simulation Objectives 1 Illustrate the steps necessary to model a system that includes elements from : mechanical, thermodynamic, fluid dynamic subsystems. 2 Introduce the notion of so-called hybrid systems. G. Ducard c 29 / 66

30 - Simulation Problem Definition Water-propelled Rocket (WPR) V(t) p(t) v(t) (absolute) h(t) m w (t) w(t) (relative) G. Ducard c 30 / 66

31 - Simulation Bottle: mass m l and volume V l Initial conditions At start, filled with water: mass of water m w (0) occupying a volume V w (0) according to: V w (0) = m w(0) ρ w The rest of the volume is filled with air: V a (0) according to: at a certain pressure p(0). V a (0) = V l V w (0), G. Ducard c 31 / 66

32 - Simulation Flight sequence: At t = 0, the nozzle is opened. 1 0 < t < t 1 : Lift force due to water jet (until there is no water anymore), 2 t 1 < t < t 2 :Thrust due to pressurized air (until the air pressure in the rocket reaches atmospheric pressure), 3 t > t 2 : Ballistic mode. G. Ducard c 32 / 66

33 - Simulation Hybrid system: Definition A system that changes its dynamic behavior depending on discrete events is referred to as a hybrid system. In this WPR example the discrete events are switching conditions : when all the water has been flushed away: m w (t) = 0 when there is no more pressurized air: m air (t) = 0 Consequence: the 3 phases (water thrust, air thrust, ballistic flight) will be modeled separately. G. Ducard c 33 / 66

34 - Simulation Assumptions 1 Only the vertical motion is modeled (WPR assumed to only move up or downward). 2 Only 2 forces are considered: - gravity - thrust (from water, air). Aerodynamic forces are neglected. 3 The expansion of the pressurized air inside the WPR is isentropic (no heat transfer, no friction considered) 4 Compared to the mass of water, the air mass is neglected. m a m w 5 The flow of the fluids through the nozzle may be modeled using Bernoulli s law (fluid assumed incompressible without friction). G. Ducard c 34 / 66

35 - Simulation Outline 1 Lecture 9 - A: Chemical Systems Example: Continuously Stirred Tank Reactor Simulation Simulation Objectives Simulation Setup Simulation Results G. Ducard c 35 / 66

36 - Simulation time t time t+dt v(t) m(t) m(t) v(t)+dv(t) dm(t) w(t) v(t) dm(t) Figure: Illustration of the momentum balance equations. The velocity of the water or air, respectively, ejected through the nozzle is w(t). This velocity is relative to the WPR and is positive when flowing in the direction indicated in the Figure. G. Ducard c 36 / 66

37 - Simulation time t time t+dt v(t) dm(t) m(t) m(t) v(t)+dv(t) w(t) v(t) dm(t) Figure: Illustration of the momentum balance equations. Momentum balance: db(t) = B(t+dt) B(t) G. Ducard c 37 / 66

38 - Simulation time t time t+dt v(t) m(t) m(t) v(t)+dv(t) dm(t) w(t) v(t) dm(t) db(t) = B(t+dt) B(t) = [m(t)(v(t)+dv(t)) dm(t)(w(t) v(t))] [m(t) + dm(t)]v(t) = m(t) dv(t) dm(t) w(t) G. Ducard c 38 / 66

39 - Simulation Newton s law db(t) = F e (t) dt = g m(t) dt Combining equations: m(t) dv(t) dm(t) w(t) = g m(t) dt m(t) dv(t) = dm(t) w(t) g m(t) dt m(t) dv(t) = dm(t) w(t) g m(t) dt dt G. Ducard c 39 / 66

40 - Simulation Water mass flow F: nozzle area [m 2 ] dm(t) dt = m = ρ F w(t) Dynamic equations vertical motion of the WPR : m(t) d dt v(t) = g m(t)+ρ F w2 (t) }{{} T w mass change of the WPR: d dt m R(t) = ρ F w(t) Now, can we find a way to express w(t)? G. Ducard c 40 / 66

41 - Simulation p(t) air p(t), velocity= 0 water nozzle area F p a,w(t) Velocity w(t) of the water in the nozzle defined by the pressure difference over the nozzle p(t) p a (Bernoulli, water considered incompressible) 1 2 ρ w2 (t)+p a = p(t) Are neglected, the increase of pressure due to: 1 gravity 2 and acceleration. G. Ducard c 41 / 66

42 - Simulation p(t) air p(t), velocity= 0 water nozzle area F p a,w(t) Velocity w(t) is obtained as: 2 w(t) = p(t) p a ρ G. Ducard c 42 / 66

43 - Simulation Description of the pressure p(t) in side the WPR The pressure p(t) is the pressure of the air inside the WPR (starts at p(0)). Water gets off the WPR the volume available for air increases the pressure p(t) decreases V(t) = V l m w(t) ρ Assuming isentropic conditions (PV κ = cst) ( ) V(0) κ p(t) = p(0) V(t) where κ = c p /c v 1.4 for air. G. Ducard c 43 / 66

44 - Simulation Outline 1 Lecture 9 - A: Chemical Systems Example: Continuously Stirred Tank Reactor Simulation Simulation Objectives Simulation Setup Simulation Results G. Ducard c 44 / 66

45 - Simulation Phase 2: Air-thrust, t 1 < t < t 2 Thrust due to the compressed air flow. 1 In phase 1: only 1 state variable is sufficient to describe quantities for both fluids in the rocket: water mass m w. 2 In phase 2: the mass of the air in the rocket is not constant anymore 2 state variables are used to describe the behavior of the gas flow (air): air mass m air and air internal energy U. Mass of air contained in the rocket? The air mass at the beginning of the air-thrust phase (t = t 1 ) is equal to the mass of air at the beginning of the water-thrust phase at t = 0. m air (t 1 ) = m air (0) G. Ducard c 45 / 66

46 - Simulation Phase 2: Air-thrust, t 1 < t < t 2 Air mass m air (t 1 ) = m air (0) Using the ideal-gas law: P(0) V(0) = n air R u T = m air M air R u T m air (0) = P(0) V(0) M air R u T = cst for t < t 1 with the universal gas constant: R u = 8.31 [J. K 1. mol 1 ], molar mass of air: M air = [kg. mol 1 ]. G. Ducard c 46 / 66

47 - Simulation Phase 2: Air-thrust, t 1 < t < t 2 Thrust due to air mass flow ( m out )? m out = d dt m air = ρ air F w(t) db(t) = B(t+dt) B(t) = m(t) dv(t) dm(t) w(t) db(t) = m(t) g dt G. Ducard c 47 / 66

48 - Simulation Neglecting the mass of air m air compared to mass of the rocket m R m air m R m(t) = m R db(t) = m R g dt = m R dv(t) dm(t) w(t) = m R dv(t) ρ air F w(t)dt w(t) Rocket dynamics equation during the air-thrust phase m R dv(t) dt = m R g +ρ air F w 2 (t) }{{} T air Remark: compare with the dynamic equation during water-thrust phase: m(t) d dt v(t) = m(t) g +ρ w F w 2 (t) G. Ducard c 48 / 66

49 - Simulation Phase 2: Air-thrust, t 1 < t < t 2 Thrust due to Pressurized Air Air thrust T air (t) = ρ air F w 2 (t) = ρ air F w(t) w(t) }{{} m air (t) = m air (t) = m 2 air (t) ρ air F m air (t) ρ air F Remark: this is a trick (to use the air mass flow m air ) in order not to have to compute explicitly the relative airflow speed w(t). G. Ducard c 49 / 66

50 - Simulation Phase 2: Air-thrust, t 1 < t < t 2 Air Mass-flow: m air (t) The air rocket is modeled as a gas receiver (see previous lecture) and the nozzle can be seen as a valve that restricts the exit area of the gas flow. p in (t) p out (t) m in (t),ϑ in (t),p in (t) m out (t),ϑ out (t),p out (t) ϑ in (t) ϑ out (t) G. Ducard c 50 / 66

51 - Simulation If we assume conditions of perfect gases: p m in (t) air (t) = c d F R ϑin (t) Ψ(p in(t),p out (t)) where Ψ(.) is approximated (air and many other gases OK) by: Ψ(p in (t),p out (t)) = In this case: 1 2 [ ] 2p out p in 1 pout p in for 2p out < p in for 2p out p in p in (t) is the pressure of air inside the rocket: P air (t), p out (t) is the pressure of atmosphere: P a, ϑ in (t) is the temperature of air in the rocket: ϑ(t) G. Ducard c 51 / 66

52 - Simulation Approximation (unrealistically large threshold: π tr = 0.9) Ψ(Π) (-) Ψ exact (solid) and approximated (dashed)(κ = 1.4), laminar part (dash-dot) Π (-) G. Ducard c 52 / 66

53 - Simulation Air Pressure P air (t) during the Air-thrust Phase See Gas Receiver example p.45 and p.46 in the script. (lecture 6) d κr { min p(t) = (t)ϑ in (t) m out (t)ϑ(t)} dt V taking m in (t) = 0, we get: with: d { mout } p(t) = κr (t)ϑ(t) dt V l V l : total volume of the rocket (bottle) ϑ(t): temperature of air in the rocket. G. Ducard c 53 / 66

54 - Simulation Air Temperature ϑ(t) during the Air-thrust Phase See Gas Receiver example p.45 and p.46 in the script. (lecture 6) d ϑ(t) R { } ϑ(t) = c p min ϑ in c p mout ϑ ( m in m out )c v ϑ dt p(t) V l c v taking m in = 0, we get: d dt ϑ(t) = = ϑ(t) R { } c p mout (t) ϑ(t)+ m out (t) c v ϑ(t) p(t) V l c v ϑ(t) R { (c p c v ) m } out (t) ϑ(t) p(t) V l c v knowing that for ideal gas R = c p c v in [J (kg K) 1 ], we finally obtain: d dt ϑ(t) = ϑ2 (t) R 2 m out (t) p(t) V l c v G. Ducard c 54 / 66

55 - Simulation Outline Simulation Objectives Simulation Setup Simulation Results 1 Lecture 9 - A: Chemical Systems Example: Continuously Stirred Tank Reactor Simulation Simulation Objectives Simulation Setup Simulation Results G. Ducard c 55 / 66

56 - Simulation Simulation Objectives Simulation Objectives Simulation Setup Simulation Results 1 What is the optimum mass of water that you should put in the rocket to reach maximum height? 2 What is the maximum reachable height? G. Ducard c 56 / 66

57 - Simulation Outline Simulation Objectives Simulation Setup Simulation Results 1 Lecture 9 - A: Chemical Systems Example: Continuously Stirred Tank Reactor Simulation Simulation Objectives Simulation Setup Simulation Results G. Ducard c 57 / 66

58 - Simulation Simulation Objectives Simulation Setup Simulation Results Simulation Setup: Water-thrust Phase Simulation of the rocket from start to "water burn out" total mass of the "rocket" time t isentropic expansion m p pressure air in plenum nozzle behavior p w relative velocity of water leaving nozzle water mass balance w m w T thrust generation thrust m T vertical motion G. Ducard c 58 / 66

59 - Simulation Simulation Setup: Phase Switch Simulation Objectives Simulation Setup Simulation Results m(t) 0 0 h t 0 t 0 time t Figure: Iterative state-event detection. G. Ducard c 59 / 66

60 - Simulation Simulation Objectives Simulation Setup Simulation Results Simulation Setup: Air-thrust Phase G. Ducard c 60 / 66

61 - Simulation Outline Simulation Objectives Simulation Setup Simulation Results 1 Lecture 9 - A: Chemical Systems Example: Continuously Stirred Tank Reactor Simulation Simulation Objectives Simulation Setup Simulation Results G. Ducard c 61 / 66

62 - Simulation Simulation Results Simulation Objectives Simulation Setup Simulation Results 20 Maximum height reached 20 Maximum speed reached Height [m] Height [m] Speed [m/s] Water content in % of WPR volume Water content in % of WPR volume Height trajectory at optimum water content speed trajectory at optimum water content Speed [m/s] Time t Time t G. Ducard c 62 / 66

63 - Simulation Simulation Results Simulation Objectives Simulation Setup Simulation Results Top two plots Maximum height and maximum speed reached during the flight of the WPR for varying initial water levels. The crosses in the top left diagram indicate measured values. Bottom two plots Height and speed trajectories for a WPR with optimal water content at start. The dashed curves are obtained when neglecting the thrust contribution of the air after water burn out. G. Ducard c 63 / 66

64 - Simulation Simulation Objectives Simulation Setup Simulation Results Discussion on the Simulation Results Qualitatively, this result can be explained by the following arguments: 1 If the WPR is filled with very little water: a large amount of energy can be stored in the pressurized air. However, this energy will not produce much thrust because of the lack of sufficient propellant. 2 If too much water is used: very little energy can be stored in the pressurized air. Moreover, the WPR will be very heavy and lifting that mass will require a large amount of the energy stored in the pressurized air. G. Ducard c 64 / 66

65 - Simulation Simulation Materials Simulation Objectives Simulation Setup Simulation Results Model Equations Complete model equations can be downloaded at http : modeling.html G. Ducard c 65 / 66

66 - Simulation Simulation Objectives Simulation Setup Simulation Results Thank you for your attention. G. Ducard c 66 / 66

Modeling and Analysis of Dynamic Systems

Modeling and Analysis of Dynamic Systems by Dr. Guillaume Ducard c Fall 2016 Institute for Dynamic Systems and Control ETH Zurich, Switzerland 1/59 Outline 1 2 Introduction Example: Continuously Stirred