Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online) DOI 10.1007/s12206-016-0149-y Numerical analysis of transient pressure variation in the condenser of a nuclear power station† Xinjun Wang1, *, Zijie Zhou1, Zhao Song1, Qiankui Lu2 and Jiafu Li2 Institute of Turbomachinery, Xi’an Jiaotong University, Xi’an, 710049, China Dong Fang Turbine Co., Ltd, Deyang, 618000, China (Manuscript Received June 2, 2015; Revised September 5, 2015; Accepted September 24, 2015) Abstract To research the characteristics of the transient variation of pressure in a nuclear power station condenser under accident condition, a mathematical model was established which simulated the cycling cooling water, heat transfer and pressure in the condenser The calculation program of transient variation characteristics was established in Fortran language The pump’s parameter, cooling line’s organization, check valve’s feature and the parameter of siphonic water-collecting well are involved in the cooling water flow’s mathematical model The initial conditions of control volume are determined by the steady state of the condenser The transient characteristics of a 1000 MW nuclear power station’s condenser and cooling water system were examined The results show that at the condition of plant-powersuspension of pump, the cooling water flow rate decreases rapidly and refluxes, then fluctuates to The variation of heat transfer coefficient in the condenser has three stages: at start it decreases sharply, then increases and decreases, and keeps constant in the end Under three conditions (design, water and summer), the condenser pressure goes up in fluctuation The time intervals between condenser’s pressure signals under three conditions are about 26.4 s, which can fulfill the requirement for safe operation of nuclear power station Keywords: Condenser of nuclear power station; Power supply halt of pump; Flow rate of cooling water; Transient heat transfer; Pressure of condenser Introduction As an important equipment in condensing turbine set, the condenser can keep stated backpressure and coagulate steam into clean water The condenser is a kind of heat exchanger which plays the role of cold source in the turbine’s thermodynamic system For security thought, three signals are settled in a condenser: signal of emergent halt of turbine, signal of emergent halt of reactor and signal of emergent halt of condenser When there is a forced outage accident in a nuclear power station, the recycle water pump shuts down As a result, the flow rate of cooling water decreases to zero gradually due to siphonage In consequence, the heat transfer coefficient and capacity of heat transmission in condenser decline and the condenser pressure goes up dramatically Ultimately, the rising of pressure in condenser triggers signals of emergent halt of the turbine and emergent halt of the reactor, which causes the shutdown of both the reactor and main valve of the turbine But there is still live steam production after the shutdown of the reactor To avoid overpressure of the second circuit, live * Corresponding author Tel.: +86 29 82660313, Fax.: +86 29 82660313 E-mail address: xjwang@mail.xjtu.edu.cn † Recommended by Associate Editor Kwang-Hyun Bang © KSME & Springer 2016 steam should be discharged through the bypass system into the condenser, which means the pressure in the condenser will boost further Consequently, if the signal of emergent halt of the condenser is triggered because of the high pressure, the bypass valve will close so the main steam cannot flow into the condenser, which means the emergency condition in nuclear power station will turn worse Therefore, the condenser in a nuclear power station must fulfill both the requirements under normal condition and emergency condition In emergency condition, a condenser is required to take in and condense the steam of high temperature and high pressure timely To determine the pressure of emergent halt of condenser signal, the reliability, safety and economy of the system should be taken in consideration If the pressure of emergent halt of reactor or emergent halt of turbine is set at a low level, the frequent fluctuation of condenser’s pressure will cause the continual halt of turbine and reactor, which will shorten the lifetime of the turbine set and the reactor If the pressure is set too high, the condenser pressure will trigger an emergent halt of the condenser signal in no time when live steam flows into the condenser through the bypass system In this situation, the requirement of a nuclear power station’s secure operation cannot be fulfilled Considering the time needed for the halt and the main steam flow, the interval between emergent halt 954 X Wang et al / Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 of turbine and emergent halt of condenser should be 12 s or longer The sudden absence of the house power is the worst situation in a nuclear power station The interval between these two signals directly and dominantly influences the station's security and is an important index parameter of condenser and cooling water system design Much research has been conducted on flow rate and the transient pressure variation at the condition of plant-powersuspension of pump in a power station [1, 2] But research on condensers has mainly focused on some aspects like flow and heat transfer on steady condition, variable working condition and dynamic performance In the 1970s, Spalding and Patankar [3] proposed the idea of a porous media model In that case, the flow in steam side of the condenser is simplified to the flow of mixture of steam and air in porous media with distributed resistance and distributed mass Hu and Zhang [4] investigated the condensate tubes that were submerged by condensing water, and proposed a new relation to calculate the heat transfer and flow characteristics in the condenser Hou et al [5] did numerical research on flow field and heat transfer characteristics in the vapor side of a condenser using the porous media model Yang et al [6] numerically simulated the flow in the vapor side in condenser of a 300MW power station using PHONEICS software Oh and Revankar [7] did some research on the surface condenser with elevation arranged tubes by experiment, and observed some characteristics of heat transfer and variable working conditions Using the overall heat transfer coefficient equation of condenser proposed by the Heat Transfer Institute (HEI), Raj [8] gave some prediction and advice on different conditions of condenser Zhu et al [9] did some experimental study and indicated the cooling water flow's influence on condenser’s heat transfer coefficient, temperature difference and pressure Electric Power Research Institute (EPRI) [10] developed a real-time simulation software (MMS) which can simulate the dynamic performance of different facilities in the unit Furthermore, this software can also evaluate the influence of difference disturbances on the operation of the unit Carcasci and Facchini [11] at the Institute of Automation and Computing of Italy developed a highly flexible computerized method which can real-time simulation on the power station Using the method of artificial neural network, Prieto et al [12] made predictions of the heat transfer coefficient and cleanness factor in the condenser of a power station cooled by sea water And the result of the prediction seems to be accurate with error less than 5% In the condition of plant-power-suspension of pump in power station, the transient heat transfer process in condenser is of changeable pressure, changeable steam flow and changeable cooling water flow Little attention has been devoted to this situation expect for Jiang and Ding [13], who studied the transient pressure variation in designed and summer conditions when cooling water lost In Ref [13], the HEI formula was used in the calculation of the heat transfer rate And the study in Ref [13] takes no account of the change of the heat transfer area caused by the reflux of the cooling water Fig Schematic of the once-through siphonic cooling system We analyzed the characteristics of cooling water flow and the transient variation of pressure in a 1000 MW nuclear power station condenser when lost house power consumption The flow of cooling water is under the effect of siphonage Different from the literature study [13], the formula by partial is used in the calculation of the heat transfer rate And the change of the heat transfer area caused by the reflux of the cooling water is taken into consideration The results of this paper may provide a new theoretical foundation for the secure operation of a nuclear power station Transient computation of cooling water flow When cycling water pump lost its power, the cooling water flow in the once-through siphonic cooling system changes The water flow can be affected by the pump, arrangement of cooling line, check valve’s specialty and the parameter of the siphonic water-collecting well 2.1 Model and governing equation of water hammer Fig shows the once-through siphonic cooling system of a 1000 MW nuclear power station The cooling water comes out from the condenser and discharges into the front pool of the siphonic water-collecting well When the water level of the front pool climbs higher than the siphon wall (as shown in the shaded part), the cooling water overflows into the back pool of the siphonic well and comes into the natural water source Water hammer, manifested as the violent changes of fluid flow rate and pressure, is a transient process which occurs when s pump starts, pump suspends or check valve closes In calculation, the siphonic well is assumed to be a pool with constant water level All the cooling lines in the condenser are equivalent to one pipe whose flux is equal to the total flux of the cooling water The sectional area of this equivalent pipe is equal to the total sectional area of all cooling lines The water hammer wave’s propagation speed and friction coefficient in 955 X Wang et al / Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 this pipe are the same as in one cooling line In this way, the whole system can be simplified to a system of “pump - inlet pipe - equivalent cooling line - outlet pipe - water well” Darcy-Weisbach’s friction expression is introduced in and the kinematic equation of the water hammer is deduced as follows: t P C+ CX A B x Fig Computational grid of water hammer ∂Ve ∂Hp f Ve2 ∂ Ve − = + + g ∂t D g ∂x g ∂x (1) where the inertial force of unit volume is expressed on the left The three parts on the right mean the pressure of unit volume of fluid, the friction resistance head of unit length and the velocity head The continuity equation of water hammer has the form: g ∂Hp ∂Ve =− ∂x a ∂t (2) Coefficient γ is introduced in the linear combination of Eqs (1) and (2): (Ve + γ a ∂Ve ∂Ve ∂Hp ∂Hp ) + + ( g + γ Ve) +γ =0 g ∂x ∂t ∂x ∂t (3) We assume that Ve(x, t) and Hp(x, t) are the solutions of Eq (3) Comparing the total derivative of Ve(x, t) and Hp(x, t) , we obtain: dx a2 = Ve + γ dt g dx = g + Ve dt λ (4) g a H i −1 − TB ⋅ (Qp pi − Qpi −1 ) − FP ⋅ Qpi −1 ⋅ Qpi −1 = H pi H i +1 + TB ⋅ (Qp pi − Qpi +1 ) + FP ⋅ Qpi +1 ⋅ Qpi +1 = H pi (5) (8) where the subscripts ( i-1, i+1 and Pi ) stand for the position A, B and P in Fig H i +1 , H i −1 , Qpi +1 and Qpi −1 respectively stands for the mass flow and pressure head of different nods a moment before Their units are m and m3.s-1 H pi (m) and Qpi ( m3.s-1) are the transient pressure and mass flow ∆x (m) is the step length of the pipe TB = The solution of Eq (4) is γ =± Hp(x, t) satisfy the differential relations on their own characteristic lines Fig shows the computational grid of water hammer basis on the characteristic line equation set (Eqs (6) and (7)) In this grid, the x axis is along the length of the pipe and y axis denotes time In view of friction resistance, the whole pipe is sectioned into n parts and the time step is ∆t = ∆x / a Generally speaking, when ∆x and ∆t minish, the result of computation will approach the real transient flow situation In this figure, AP and BP (marked as C+ and C-) whose slope are 1/a and -1/a, are characteristic lines in x-t plane Discretize the water hammer’s characteristic line equation by time, and then we have the general expression of discrete equations: a f ∆x and FP = Ag DgA2 are the calculation factors For a certain cooling line system, the values of TB and FP not change over t and x Once the flow state before ∆t is known, they can be easily worked out by Eq (8) 2.2 Numerical condition Substituting γ into Eqs (3) and (4), then we have: f dVe g dHp dt + a dt + D ⋅ Ve ⋅ Ve = Along C ： dx = a + Ve ≈ a dt (6) f dVe g dHp − + ⋅ Ve ⋅ Ve = ― a dt 2D Along C ： dt dx = −(a + Ve) ≈ − a dt (7) + Eqs (6) and (7) are the characteristic line equation sets and dx / dt is the expression of characteristic line Ve(x, t) and In the computation process of water hammer using the characteristic line equations, the initial conditions and the boundary conditions of cooling water are needed Initial conditions are parameters of steady state of the cooling water Boundary conditions consist of (1) the suspension condition of pump, (2) the condition of cascaded pipeline of cooling water, and (3) the constant pressure of the pool For detailed expressions, refer to Ref [2] 2.3 Verification of the program The self-written program is checked by the example "water hammer caused by suspension of pump in a valveless pipe" in 956 X Wang et al / Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 Table Condition of the check calculation Parameter Value Unit Rotating inertia moment of pump 636.5 N·m2 Rated flow of pump 0.912 m3/s Rated lift of pump 67.1 m Rated torque of pump 1283.9 N·m Rated speed of pump 1760 r/p Number of pipe \ m Diameter of pipe 0.813 Fraction factor of pipe \ Water hammer wave velocity 860 m/s Height of pump m Height of pool 67.1 m Calculation time duration 30 s 120 100 H/m 80 β(result of literature[1]) β(result of check calculation) v(result of literature[1]) v(result of check calculation) h(result of literature[1]) h(result of check calculation) 60 40 -1 20 -2 10 15 20 25 30 v,β 140 -3 time/s Fig Comparison of check calculation result and the result of Ref [1] Ref [1] The calculation condition is briefly listed in Table For more details, see Ref [1] Fig shows the comparison calculation in Ref [1] and the results of this paper The curves in Fig express the pressure head, non-dimensional mass flow and non-dimensional torque From this comparison we see that the maximal pressure head in this paper is 97 m, while that in Ref [1] is 96.7 m The max reflux in this paper is 0.98 m3/s, while that in Ref [1] is m3/s After comparing these two results, a conclusion is drawn that the self-written calculation program is reliable enough to get a precise result Transient pressure calculation of condenser In a power station’s surface condenser, the cooling water is insulated from the steam by the wall of cooling lines The space in the condenser can be divided into two sides: the steam side and the water side The space of steam side in condenser is invariant and can be divided into three parts: gaseous phase part, air part and hot well part Though there is no obvious bound between gaseous phase part and air part, the interfaces between the hot well part and the gaseous part and the interface between the hot well part and the air part are vivid The water side in the condenser is of two parts: cooling lines and cooling water Cooling lines can be divided into two parts: exposed part and nonexposed part (nonexposed part is located in the condensed water) The exposed part and the nonexposed part convert with the change of quantity of condensed water 3.1 Calculation model and governing equation It’s a complex procedure of flow and heat transfer when there is a plant-power-suspension of pump in a power station The steam discharged into the condenser cannot be condensed completely so that the temperature and pressure in the condenser will rise up The steam then fills in the steam side and a heat transfer and condensation procedure with changeable pressure and mass happens The pressure and mass flow of the cooling water decrease to zero gradually With the rise of cooling water’s level, the cooling water lines become submerged To calculate the transient change of pressure in the condenser, we set the gaseous side of condenser and exhausted casing of turbine (except for the condensed water) as control volume Assume that: (1) The gaseous phase of the condenser is in thermodynamic equilibrium; (2) The steam discharged into condenser from bypass system is of high pressure, so that the mass flow of discharged steam cannot be influenced by the pressure fluctuation of condenser; (3) The mass flow and flow rate of cooling water in condenser are homogeneous in calculation; (4) Take no account of chemical filling water and flash of condensation; (5) In every time node, the heat transfer is steady and the changes between neighboring time nodes are step changes The continuity equation of control volume open system can be written as: ∑ Gɺ v , in where dGv − Gɺ = , dt ∑ Gɺ v ,in (9) denotes the mass flow of total steam (includ- ing steam from the bypass system and the turbine’s exhaust steam) into the control volume Gɺ is the total condensation rate Gv is the steam mass in the control volume And t denotes time The energy equation of the control volume's vapor filling procedure becomes (ignore the kinetic energy and potential energy of the steam ): −Qɺ + ∑（Gɺ ɺ − Gcon hl , s v , in hv , in） = dU OPS , dt (10) where Qɺ denotes the capacity of heat transmission between the control volume and outside When the control volume 957 X Wang et al / Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 releases heat to the outside, Qɺ is positive Otherwise, Qɺ is negative （Gɺ v ,in hv ,in）denotes the energy which is brought ∑ along with the steam into the control volume hv ,in is the specific enthalpy of the steam in control volume Gɺ hl , s is the energy which is brought away by the condensed water hl , s is the specific enthalpy of saturated water under the pressure of condenser U OPS is the total energy of the control volume The rate of internal energy’s change in control volume can be written in the form: dU OPS dGv du = ⋅ uv + v ⋅ Gv dt dt dt (11) The capacity of heat transmission Qɺ has two different sources: One, the capacity of the heat transmission between steam and water; two, from the heat transmission between steam , cooling line, condenser casing, and the heat transmission between casing and outside We then write the relation between the capacity of heat transmission and the condensation rate: Qɺ = Gɺ (hv − hl , s ) (12) Discretize Eqs (9)-(12) by time, and we have the governing equations of the transient pressure change of condenser: t +∆t t t ɺ t + ∆t Gv − Gv Gɺ vt ,+∆ in − Gcon = ∆t t +∆t t ɺ t +∆t + （Gɺ t +∆t ⋅ ht +∆t）− Gɺ t +∆t ⋅ ht +∆t = U ops − U ops −Q v , in v , in l ,s ∆t U t +∆t − U t t +∆t t t +∆t t G − G u − u ops v v ops = v × uvt +∆t + v × Gvt +∆t ∆t ∆t ∆t ɺ t +∆t ɺ t +∆t t +∆t t = Gcon (hv − hlt,+∆ s ) Q ∑ ∑ use the formula recommended by Gnielinski: ( f / 8)( Rew − 1000) Prw λw d × + 12.7( f / 8)1/ ( Pr 2/3 − 1) , i w αw = λ w 4.36 × , Rew < 2300 di (15) where aw is the forced convection heat transfer coefficient in tube λw is the heat conductivity coefficient of the cooling water di denotes the inner diameter of the cooling line Rew denotes the Reynolds number of the cooling water flow and Prw is the Prandtl number of the cooling water For the outside surface of the cooling line, we use the correction heat transfer coefficient of film condensation in Ref [16]: (16) where α v is the condensation heat transfer coefficient in the steam side Π expresses the influence of air flow’s shear stress on the water film outside the tube α n is the film condensation heat transfer coefficient of the horizontal tube Nu is the Nusselt number corresponding to α n Z is the number of passes S is the correction factor of cooling lines organization ε is the correction factor of air content The specific expression of each coefficient can be found in Refs [14, 16] When the flow rate of cooling water is less than 0.9 m/s, the overall heat transfer coefficient of condenser can be expressed as: K= (13) Rew ≥ 2300 , do d + ⋅ ln o + α w d i 2λm di α v (17) where λm ( W/m.K )is the heat conductivity of cooling line d o (m) denotes the outside diameter of the cooling line pipe 3.2 Calculation of heat transfer coefficient When the flow rate of cooling water is higher than 0.9 m/s, HEI [8] standard can be used to calculate the overall heat transfer coefficient of condenser: K = K β c βt β m , (14) where K0 is the basic heat transfer coefficient βc is the correction factor of cooling lines' cleanliness βt is the correction factor of temperature β m is the correction factor of cooling lines' material and wall thickness The specific value of each factor can be found in Ref [15] When the flow rate of cooling water is less than 0.9 m/s, the overall heat transfer coefficient can be calculated using experimental correlations For forced heat convection in tube, we 3.3 Calculation of the temperature of the cooling water and the capacity of heat transmission Fig shows the calculation grid of cooling water’s flow and heat transfer The cooling line’s wall is shown as the shaded portions The grid is of four sections: inlet Sec (I), heat transfer Sec (W) and outlet Sec (O), and grid M The first three sections of grid trace the position and temperature of cooling water so that the whole pipe’s capacity of heat transfer can be calculated no matter if backflow happens Grid M is used for comparison General calculation steps are: (1) Calculate the cooling water node’s position at t and t + ∆t separately according to the known change regularity of cooling water’s flow rate; (2) Confirm the grid nodes that are in the cooling line at ∆t 958 X Wang et al / Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 partial should be corrected When corrected, the heat transfer rate, pressure and temperature of condenser should be the same as the steady state of calculation The correction factor is equal to the heat transfer rate which was calculated from the formula by partial dividing the heat transfer rate calculated from overall: Fig Calculation grid of cooling water's flow and heat transfer period by their position coordinates Only these nodes participate in the heat transfer process (3) Calculate cooling water’s temperature and heat transfer capacity by HEI formula or the formula by partial The I, W, O grid of cooling water should be taken into consideration (4) Acquire every node’s temperature at the moment of t + ∆t Use linear interpolation strategy to calculate M grid’s temperature at the moment of t + ∆t Prepare for next calculation The logarithmic mean temperature difference is mainly adopted in the calculation of steady calculation of heat transfer in condenser In this study, the length of pipe of each grid is small enough and the temperature variation is not remarkable In the calculation of cooling water’s temperature, the temperature difference between cooling water and steam is used for the heat transfer temperature difference Adopting implicit format time discretization on grid node WN we have the equation of heat balance: QɺWn = t cWn mWn (TWt +∆ − TWt n ) n ∆t t = KWt +∆ AWn (Tvt +∆t − TWt n ) , n (18) where c denotes specific heat at constant pressure; m is the mass of cooling water; T is the temperature; ∆t denotes the time step length; A is the area of heat transfer In the subscript, Wn is the number of segment and v means steam From Eq (18), we have the temperature calculation equation of cooling water at the moment of t + ∆t : t TWt +∆ = n t KWt +∆ AWn Tvt +∆t + cWn mWn TWt n n t KWt +∆ AWn + cWn mWn n (19) Calculating all the nodes in grid W according to Eqs (18) and (19), then we have the cooling water’s temperature of grid W and the heat transfer rate Qɺ Wn of each segment of cooling water We have to use iteration in the calculation of temperature and heat transfer rate because of using implicit format After we know the temperature rise and heat transfer rate of all the nodes, the overall heat transfer rate of cooling water and condenser can be calculated: Qɺ = ∑ Qɺ Wn (20) Strictly speaking, the heat transfer rate calculated from HEI equations and the formula by partial is aimed at the whole condenser Here, the result calculated from the formula by Qɺ HEI ,1 / Qɺ HEI fac = Qɺ SEP ,1 / Qɺ SET rate > 0.9 m/s rate < 0.9 m/s (21) where Qɺ HEI ,1 is section’s heat transfer rate calculated from HEI equations Qɺ SEP ,1 is section’s heat transfer rate calculated from the formula by partial Qɺ HEI is the overall heat transfer rate calculated from HEI equations Qɺ SEP is the overall heat transfer rate calculated from the formula by partial fac denotes the correction factor The corrected overall heat transfer rate of cooling water and condenser becomes: Qɺ1 = Qɺ / fac (22) The heat storage rate of the metal wall of cooling water line pipe is: dT Qɺ = mm cm m , dt (23) where mm expresses the mass of the wall of cooling water line pipe cm is the specific heat of metal wall of tube and Tm is the mean temperature of the cooling water line pipe The heat absorptivity of the condenser’s shell can be represented as: dT Qɺ = mshell cshell shell , dt (24) where mshell is the mass of condenser’s shell cshell is the specific heat of metal shell Tshell is the temperature of the shell Assume that the temperature of the condenser shell is always equal to the average temperature of the two sides of wall The shell’s radiation heat transfer rate to outside is: T + 273.15 Qɺ = εσ ( shell ) ⋅ Ashell , 100 (25) where ε is the blackness or emissivity which is set to 0.8 σ is the radiation coefficient of black body, set to 5.67 Ashell is the radiation area of the condenser In the control volume, the total heat transfer rate between steam and outside is: Qɺ = Qɺ1 + Qɺ + Qɺ + Qɺ (26) 959 X Wang et al / Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 3.4 Initial conditions The initial conditions of vapor side in the condenser are the same as three working conditions: designed, winter and summer working condition The initial condition of water side in condenser is the steady status of cooling water In steady working status, the temperature of all nodes of inlet Sec I is equal to the water source’s temperature The temperature of all nodes of outlet Sec O is equal to the cooling water’s temperature at the outlet of condenser The cooling water’s temperature of grid M accords exponential distribution The calculation expression of initial temperature of grid W is [15]: , (27) where x is the distance from the inlet opening of the cooling water ∆Tv , w, x is the temperature difference between steam and cooling water at x ∆Tv , w, in denotes the temperature difference between steam and cooling water at the inlet opening Ax is the heat transfer area from inlet opening to x K x is the general heat transfer from inlet opening to x qw is he mass flow of cooling water and cw is the specific heat of the cooling water Results and analysis The study is based on a 1000 MW nuclear power station set’s cycle cooling water system When 85% of the main steam is discharged into the condenser through bypass system and both the cycle cooling water and the attemperation water are lost, the transient variation of temperature in condenser is focused on The study was conducted in three working conditions: designed, winter and summer Fig Calculation process 2.5 2.0 -1 K A − x x qw cw Flow Rate/m·s ∆Tv , w, x = ∆Tv , w,in e 1.5 1.0 0.5 0.0 -0.5 -1.0 10 15 20 25 30 35 40 45 50 Time/s Fig Cooling water’s flow rate variation 4.1 The Program and process of calculation The self-written program based on FORTRAN language is applied to calculate the transient variation of cooling water and condenser’s pressure The basic calculation process is shown in Fig 4.2 Characteristics of cooling water flow In the 1000 MW power station set, there is a check valve at the outlet of cycle cooling water pump which is controlled by an electric actuator When there is plant-power-suspension of pump, the check valve is closed due to elastic force and gravity In that way, the cooling water can flow backwards in a while When the electric controlled check valve is closed linearly, the change of cooling water’s flow rate is shown in Fig The close duration time of the check valve is 45 s The variation of flow rate is of stages: at 0-18.1 s, the flow rate decreases to zero almost linearly; at 18.1-45.0 s, the cooling water flows backwards; the flow rate decreases to zero again at 45.0 s and then fluctuates; in the end, the cooling water stops flowing Fig Schematic of the cycle water system in a nuclear power station: (a) reactor; (b) steam generator; (c) bypass valve; (d) live steam valve; (e) steam turbine; (f) electric generator; (g) condenser; (h) pump of cycling water; (i) check valve; (j) pump of nuclear loop 4.3 The transient characteristics of pressure variation in condenser Fig is a schematic of the cycle water system in a nuclear 960 X Wang et al / Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 Table Condition of the condenser’s calculation Working condition Designed Summer Winter Live steam pressure (MPa) 6.5 6.5 6.55 Live steam specific enthalpy(kJ/kg) 2771.2 2771.2 2770.6 Turbine exhaust steam mass flow (kg/s) 863.3 880 839.2 Exhaust steam specific enthalpy (kJ/kg) 2319.3 2365.9 2293.7 Condenser pressure (designed) (kPa) 5.1 8.1 2.68 Attenuation rate of exhaust steam mass flow (kg/s) 4000 4000 4000 Steam mass flow of bypass (kg/s) 720 720 720 Heat transfer area of condenser (m2) 72050 72050 72050 Flow path of condenser 1 Material of cooling line Ti Ti Ti External diameter of cooling line (mm) 25 25 25 Temperature of cooling water (°C) 20.5 29.6 6.0 Flow rate of cooling water (m/s) 2.38 2.38 2.38 Volume of vapor phrase in condenser (m3) 5800 5800 5800 Volume of exhaust casing (m3) 600×2 600×2 600×2 Live steam valve close duration (s) 0.5 0.5 0.5 Bypass steam valve open duration (s) 2.5 2.5 2.5 Bypass steam close duration (s) 5 Emergent halt of turbine (kPa) 20 20 20 Emergent halt of reactor (kPa) 20 20 20 Emergent halt of condenser (kPa) 50 50 50 Steam mass flow/kg·s -1 1000 800 600 Summer Designed Winter 400 200 0 10 15 20 25 30 35 40 45 50 Time/s Heat transfer coefficient/W·m-2· ℃-1 (a) Steam mass flow variation 4000 Summer Designed Winter 3000 2000 1000 0 10 15 20 25 30 35 40 45 50 Time/s Heat transfer capacity/GW (b) Heat transfer coefficient variation 3.0 Summer Designed Winter 2.5 2.0 1.5 1.0 0.5 0.0 10 15 20 25 30 35 40 45 50 Time/s (c) Heat transfer capacity variation 0.08 Summer Designed Winter 0.07 Pressure/MPa power plant Table shows the calculation condition of the condenser The turbine exhaust steam mass flow is the mass flow that is designed in normal working condition The attenuation rate of exhaust steam means the decrease rate of turbine's exhaust steam mass flow when power is lost suddenly The steam mass flow of bypass is the steam mass flow through the bypass system to the condenser when power lost The bypass valve opens or closes linearly The variation characteristics of steam mass flow, heat transfer coefficient, heat transfer capacity, and pressure in condenser are shown in Fig Fig 8(a) shows the variation curve of steam mass flow From this we can see that the change of steam mass flow in designed, winter and summer conditions are almost the same When there is plant-power-suspension of the pump, the turbine works as usual and the exhaust steam mass flow is the 0.06 0.05 0.04 0.03 0.02 0.01 0.00 10 15 20 25 30 35 40 45 50 Time/s (d) Variation of condenser's pressure Fig Variation of condenser's parameter same as usual When the condenser pressure reaches 20 kPa and triggers an emergent halt of the turbine signal, the live steam valve closes so that the mass flow of steam that is dis- X Wang et al / Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 charged into the condenser decreases sharply When the bypass valve opens, the steam entering into the condenser through bypass system increases The emergent halt of condenser signal and emergent halt of reactor signal are triggered when the condenser’s pressure reaches 50 kPa Then the steam mass sharply decreases to zero again Fig 8(b) shows the variation heat transfer coefficient The changes of heat transfer coefficient in three working conditions are same too The variation is of three stages: first, the pipe’s convection heat transfer coefficient and the overall heat transfer coefficient decrease sharply and touch bottom when the cooling water’s flow rate is zero and the steam mass flow is minimum Second, the cooling water flows back The pipe’s convection heat transfer coefficient and the overall heat transfer coefficient increase first and then decrease Third, the heat transfer coefficient touches bottom and keeps constant while the cooling water fluctuates at and the steam mass flow is The capacity of heat transfer of the condenser is related to the heat transfer coefficient, steam mass flow and the temperature difference (the heat transfer area is fixed) Based on the change characteristics of these three parameters, the change of heat transfer capacity can be calculated and Fig 8(c) shows this change The pressure variation of the condenser, which is affected by cooling water flow rate, heat transfer coefficient, heat transfer capacity, steam mass flow and temperature difference, is shown in Fig 8(d) In three working conditions, the condenser pressure increases in fluctuation At the start of plantpower-suspension of the pump, the steam mass flow keeps invariant while the condenser pressure increases gradually and reaches the peak value in about 17 s Soon afterwards, the main steam valve closes and the bypass valve opens The steam flow rate increases and decreases sharply At the same time, the cooling water is refluxing As a result, the condenser pressure decreases a little and then increases quickly At 20 s, it peaks for the second time From 20 s to 45 s, the bypass steam mass flow is constant and the cooling water refluxes while the heat transfer coefficient and capacity increase and then decrease As a consequence, the condenser pressure decreases and then increases At about 45 s, the condenser pressure reaches the third peak value At the end, the steam mass flow that enters into condenser decreases to zero rapidly and the condenser pressure starts to decrease because of the emergent halt of the condenser signal The time when condenser pressure reaches the third peak value is later than the emergent halt of the condenser signal-50 kPa because the reactor still generates live steam while stopping From Fig 8(d), we can conclude that when there is plantpower-suspension of pump, the condenser pressure increases to 20 kPa and triggers an emergent halt of the turbine signal in some time In summer, designed and winter condition, it needs 15.7 s, 16.7 s and 17.0 s severally The pressure of the condenser increases to 50 kPa and triggers an emergent halt of the condenser signal in some time In summer, designed and winter condition, it needs 42.1 s, 43.1 s and 43.4 s severally In 961 these three conditions, the time interval between these two signals is 26.4 s, 26.4 s and 26.3 s In conclusion, the time intervals can fulfill the operation requirements of the nuclear power station Conclusions The calculation model of cooling water flow and transient pressure variation of condenser has been established using the self–written calculation program of water hammer in oncethrough siphonage cooling water system We focused on the change of cooling water flow, heat transfer and pressure in a 1000MW nuclear power station in designed, winter and summer working conditions Based on the analytical and mathematical investigation, the following conclusions may be drawn: (1) When there is plant-power-suspension of pump, the cooling water flow rate decreases sharply and refluxes, and then it fluctuates to zero (2) Corresponding to the change of cooling water flow rate and steam mass flow, the variation of the condenser’s heat transfer factor can be divided into three stages: decreases sharply, increases and then decreases, keeps constant (3) In a 1000 MW power station, if the electric controlled valve closes linearly in 45 s, the time interval between condenser-pressure-signal of emergent halt of turbines and that of condensers is more than 12 s, which can fulfill the operation requirements of the nuclear power station no matter if the work condition is designed, winter or summer (4) The calculation model of cooling water flow and transient pressure variation of condenser in once-through siphonage cooling water system can be used in the accident scenario in a nuclear power plant The self-written program may provide an analysis method and tool for the design of a nuclear power plant's condensation system Nomenclature -a A c di D f fac g Gɺ h H Hp K K0 m Nu Prw : Spread rate of water hammer, m/s : Heat transfer area, m2 : Constant pressure specific heat, kJ/(kg.K) : Inner diameter of cooling line, m : External diameter of cooling pipe, m : Diameter of pipe, m : Resistance coefficient of pipe : Correct factor of heat transfer rate : Gravitational acceleration, m/s2 : Steam mass flow, kg/s : Specific enthalpy, kJ/kg : Pressure head of cooling water, m : Lift of water pump, m : Overall heat transfer coefficient of condenser, W/(m2.K) : Basic heat transfer coefficient of HEI, W/(m2.K) : Mass of cooling water, kg : Nusselt number corresponding to : Prandtl number of cooling water 962 Qp Qɺ Rew S t ∆t T u U Ve x ∆x Z αn αv αw βc : βm βt ε0 Π λ X Wang et al / Journal of Mechanical Science and Technology 30 (2) (2016) 953~962 : Volume flow rate of cooling water, m3/s : Heat transfer capacity between control volume and outside, kW : Reynolds number of cooling water : Correct number factor of cooling line’s organization : Time, s : Time step length , s : Temperature , K : Internal energy, kJ/kg : Total energy of control volume, kJ : Flow rate of cooling water, m/s : Flow direction of cooling water : Step length of calculation pipe Flow path of condenser : Film condensation heat transfer coefficient on horizontal circular tube, W/(m2.K) : Condensation heat transfer coefficient of vapor phase, W/(m2.K) : Forced convection heat transfer coefficient inside tube, W/(m2.K) : Correction factor of cleanness of tube : Correction factor of wall thick and material of tube : Correction factor of cleanness of tube : Correction factor of air content : Influence factor of air flow’s sheer stress on the water film outside the cooling piping : Heat conductivity coefficient, W/(m2.K) Subscripts i in l m o ops s shell v w : Heat transfer coefficient between condenser and cooling water (calculated) : Heat transfer coefficient between condenser and cooling water (corrected) : Heat transfer rate between condenser and cooling piping’s wall : Heat absorptivity of condenser’s shell : Radiation heat transfer rate between condenser’s shell and outside : Steam condensation in control volume : Internal : Flow into : Condensation water : Cooling line : External : Control volume : Saturated water : Condenser shell : Steam : Cooling water References [1] N C Jiang and X H Wang, Water hammer and protection, China Building Industry Press, Beijing, China (1993) [2] Z X Liu and G L Liu, Water hammer protection of pump stations, China Water & Power Press, 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Heat transfer, Higher Education Press, Beijing, China (2005) [15] Heat exchange institute, HEI standards for steam surface condensers, 9th Ed., Heat Exchange Institute Inc., Cleveland, OH (1995) [16] Z C Zhang, Condenser in large scale power plant, China Machine Press, Beijing, China (1993) Xinjun Wang, Ph.D., is an Assistant professor, Institute of Turbomachinery, School of Energy and Power Engineering, Xi’an Jiaotong University He has been mainly engaged in the research of aerodynamics and two-phases flow in turbomachinery ... situation in a nuclear power station The interval between these two signals directly and dominantly influences the station' s security and is an important index parameter of condenser and cooling... this software can also evaluate the influence of difference disturbances on the operation of the unit Carcasci and Facchini [11] at the Institute of Automation and Computing of Italy developed a. .. be the same as the steady state of calculation The correction factor is equal to the heat transfer rate which was calculated from the formula by partial dividing the heat transfer rate calculated