PROCEEDINGS, Thrty-Ffth Workshop on Geothermal Reservor Engneerng Stanford Unversty, Stanford, Calforna, February 1-3, 010 SGP-TR-188 GeoSteamNet:. STEAM FLOW SIMULATION IN A PIPELINE Mahendra P. Verma and Vctor Arellano G. Geoterma, Insttuto de Investgacones Eléctrcas Av. Reforma 113, Col. Palmra Cuernavaca, Morelos, 6490, Méxco e-mal: mahendra@e.org.mx ABSTRACT A computer program s developed to smulate steam flow n a ppelne as a part of GeoSteamNet: a computer package for the smulaton of steam flow n a geothermal power plant network. The flud movement s governed by the followng basc prncples: the conservaton of mass, the lnear momentum prncple (Newton s second law or Naver Stokes equatons) and the frst and second laws of thermodynamcs. The second law of thermodynamcs defnes the drecton of a spontaneous process, whch s ndrectly valdated n the algorthm as steam flows from hgh to low pressure and heat flows from hgh to low temperature. The nonlnear equatons are solved wth the Newton-Raphson method. A comparatve study on the varaton of temperature, pressure and heat loss n a ppelne of length 1000 m, nner dameter 0.3 m and thckness 0.005 m s presented. Three cases are dscussed: a) no conducton-convecton heat loss, b) an nsulaton of 0.05 m thckness on the ppelne and c) maxmum heat loss (.e. no nsulaton). The change n pressure s same n the three cases whereas there s apprecable temperature drop even n the case a. Smlarly, there s 36% densty change n the case b, whch s a restrcton to use the Bernoull s equaton for steam flow smulaton. INTRODUCTION Knowledge of numercal smulaton of steam flow n a ppelne network of geothermal system s vtal for ratonalzaton and optmzaton of steam used for electrcal energy generaton (Ruíz et al. 010). Presently, we are workng on two mportant aspects of the project: a) thermodynamc data of water and b) approprate algorthm for steam flow n a ppelne. The second aspect wll be dscussed here. The flud flow s mostly analyzed by two equatons: mass-balance (contnuty equaton) and momentum balance (Newton s second law of moton or Naver Stokes equatons) n stuatons where the flud may be treated as ncompressble and temperature dfferences are small (Mazumdar, 009). Bhave and Gupta (006) presented a comprehensve textbook on the analyss of water dstrbuton n a muncpal network. The water flow n a ppelne network s successfully modeled wth the Bernoull s theorem and Hardy Cross method. In the crcumstances when flow s compressble (densty s not constant), or occurrence of heat flux (temperature s not constant), there s need of at least one more equaton: energy balance. Smth and van Ness (1975) presented the dervaton of all the three equatons for flud flow. In ths artcle an algorthm s developed to solve numercally the three equatons: mass, momentum and energy balance for steam flow n a ppelne. An example s presented for steam flow n a ppelne of 1000 m long for three cases: a) no heat loss, b) heat loss for gven characterstcs of ppe and nsulaton of t and c) maxmum heat loss (.e. no nsulaton). FUNDAMENTAL EQUATIONS The movement of flud n a system s governed by the followng basc prncples: conservaton of mass, the lnear momentum prncple (Newton s second law or Naver Stokes equatons) and the frst and second laws of thermodynamcs (smth and Van Ness, 1975). The second law of thermodynamcs defnes the drecton of a spontaneous process. In the ppelne network of geothermal power plant the steam flows from hgh to low pressure and heat flows from hgh to low temperature. Thus the second law of thermodynamcs s ndrectly valdated and wll not be consdered here. Majumdar (1999) developed a general purpose computer program Generalzed Flud System Smulaton Program (GFSSP) to compute pressure and flow dstrbuton n a complex flud network
dl m -1, T -1, P -1 m, T, P.. Q Z -1 Z r 1 r r 3 Fgure. 1. Schematc dagram of th control volume element of a ppelne. The steam flow rate at the node -1 and are and, respectvely. Z s elevaton. T and P represent temperature and pressure, respectvely. The cross-sectonal vew of the element shows the postve heat flux Q. r 1, r and r 3 are rad of nner and outer part of the ppelne, and outer part of the nsulaton over t, respectvely. ncludng unsteady state and angular flow. In the geothermal power plant we are nterested n undrectonal steady state steam flow. The followng equatons wll be consdered for steam flow n a ppelne (Smth and van Ness, 1975): Contnuty Equaton The contnuty equaton (conservaton of mass) for steady state flow s ρ (1) u r = 0 where ρ s densty and u r s velocty. Fgure 1 shows a schematc dagram of th control volume element between nodes -1 and. The fnte dfference dscretzaton (Patanker, 1980) of contnuty equaton s expressed as ρ u () = ρ 1u 1 The subscrpt and -1 represent the values at the respect node. Conservaton of Energy The equaton of the conservaton of energy s expressed as u H + + gz = Q W s (3) where Q s the amount of heat per unt mass gven to the element from surroundngs. W s s shaft work per unt mass. H s enthalpy per unt mass and Z s the elevaton from the reference datum lne. Fgure 1 also presents a cross-sectonal vew of ppelne. The rate of heat transfer to the control volume element from the surroundngs s gven by H T = 1 hnr1 dl( Tn Tout ) ( r r ) ln( r r ) π (4) ln 1 3 1 + + + k k h r A A out 3 Where r 1, r, and r 3 are rad as shown n Fgure 1. k A and k B are thermal conductvtes of ppelne and nsulaton over t, respectvely. h n s the convectve heat transfer coeffcent between steam and nner part of the ppelne. Smlarly, h out s the convectve heat transfer coeffcent between outer part of nsulaton and surroundng ar. T n and T out are the temperature of nner steam and outer ar, respectvely. We are nterested n the steady state flow. So, the heat transferred to the control element wll be transferred to the nflowng flud. Thus the heat added (gven) to per unt mass of nflowng flud s H = T dl Q 1 + m& u The dscretzaton of energy equaton s u u 1 H H 1 + + g 1 = Conservaton of Lnear Momentum ( Z Z ) Q (5) (6) The conservaton of lnear momentum may be wrtten as VdP + udu + gdz + df = 0 (7) For both lamnar and turbulent flow the energy loss due to frcton s expressed wth the Fannng equaton
Table 1: Data used for the present modelng of steam flow n a ppelne. Parameter Value Ppelne Length (m) 1000.0 Inner dameter (m) 0.3 Thckness (m) 0.005 Thermal conductvty (W/m K) 80. Roughness (m) x10-7 Temperature (K) 450 445 440 435 No Heat Loss Gven Data Maxmum Heat loss a Insulaton Thckness (m) 0.05 Thermal conductvty (W/m K) 0.043 Convectve heat transfer coeffcent Steam and ppelne (W/m K) 30.0 Insulaton and ar (W/m K) 6.0 Inflow saturated steam Temperature (K) 450.0 Mass flow rate (kg/s) 10.0 Ar temperature (K) 300.0 Horzontal ppelne (Z=0) Pressure (Pa) 430 1000000 900000 800000 700000 600000 500000 3.0 b c fu df = dl (8) D Thus the dscretzaton of momentum equaton s % Energy loss.0 1.0 1 1 + ρ ρ 1 u 1 ( p p ) + + g( Z Z ) 1 u fuu + D 1 dl = 0 1 The dealzatons mposed n the dervaton of these equatons are descrbed n the Chapter 10 of book by Smth and van Ness (1975). A comprehensve and systematc numercal soluton approach s adapted from Patanker (1980) and Majumdar (1999). The system of nonlnear equatons s solved wth Newton- Raphson method. PROGRAM DESCRIPTION The computer program, PpeCalc s wrtten n Vsual Basc 6.0. The thermodynamc data of water are calculated from an ActveX control, SteamTablesGrd (Verma, 010) nstead of ActveX component, SteamTables (Verma, 003). (9) 0.0 0 00 400 600 800 1000 Dstance along ppelne (m) Fgure.. Calculated values of temperature, pressure and energy loss for three cases: a) no conducton-convecton heat loss, b) an nsulaton of 0.05 m thckness on the ppelne and c) maxmum heat loss (.e. no nsulaton). Densty (kg/ m 3 ) 6 5 4 3 1 0 0 00 400 600 800 1000 Dstance along ppelne (m) Fgure. 3. Varaton of densty for case b, when there s nsulaton over the ppelne. The change n densty s 36% whch s a restrcton to use the Bernoull s equaton for steam flow smulaton.
A structured varable s defned as ppe, whch stores all the nput and calculated parameters n t. The advantage of ths approach s that t s straght forward to create ActveX control. Our fnal goal s to create ActveX controls for every component of a ppelne network of geothermal system and a graphc user nterface. Ths way the program wll be a general purpose computer code for analyzng steady state flow n any geothermal ppelne network. AN EXAMPLE To llustrate the applcablty of PpeCalc an example s presented here. A horzontal ppelne of 1000 m s consdered. All the nput parameters are gven n Table 1. The ppelne s dvded nto 100 elements (.e. the length of each segment s 10.0 m). We performed prelmnary calculatons for the segment length of 1.0, 10.0 and 100.0 m. The results were n agreement for the segment length 1.0 and 10.0 m. A small segment length ncreases the accuracy n the calculated values, but t also ncreases the executon tme. So, one has to perform always some prelmnary calculaton to optmze the values of dfferent nput parameters accordng to confdence lmts of ther measured data. Ths can speed up the further calculatons to obtan relable results. Fgure shows the varaton of temperature, pressure and energy loss along the ppelne for three cases: a) no conducton-convecton heat loss, b) an nsulaton of 0.05 m thckness on the ppelne wth parameters gven n Table 1 and c) maxmum heat loss (.e. no nsulaton). The decrease n temperature s hghest for case c. There s a formaton of lqud water from the pont entrance of steam nto the ppelne and the condtons of temperature and pressure are along the saturaton curve. There s no formaton of lqud water n cases a and b and the system s n the superheated steam regon. The velocty of steam flow s approxmately 30 m/s. It means that the steam flows from one end to other wthn 35 s. Fgure c shows the loss energy for the three cases. It can be observed that there s about 3% energy loss wthn 35 s when there s no nsulaton on the ppelne (case c). One more pont to be emphaszed s that there s substantal decrease n temperature (11 K) even when there s no heat loss (case a). Ths decrease n temperature s assocated wth the expanson of vapor durng ts flow through the ppelne. Fgure 3 shows the varaton n the densty of steam for case b. It s 36 %. It means that the Bernoull s equaton has lmtatons to model steam flow n a ppelne. CONCLUSIONS The program PpeCalc s wrtten n Vsual Basc 6.0. The algorthm s based on the conservaton of mass, the lnear momentum prncple (Newton s second law or Naver Stokes equatons) and the frst law of thermodynamcs. The second law of thermodynamcs defnes the drecton of a spontaneous process. In the ppelne network of geothermal power plant the steam flows from hgh to low pressure and heat flows from hgh to low temperature. Thus the second law of thermodynamcs s ndrectly valdated. The nonlnear equatons are solved wth the Newton- Raphson method. The results obtaned from a numercal smulaton of steam flow n a ppelne for the three cases: a) no conducton-convecton heat loss, b) an nsulaton of 0.05 m thckness on the ppelne and c) maxmum heat loss (.e. no nsulaton) may be stated as: There s decrease n the temperature and pressure of steam along the ppelne even when there s no heat loss. It s assocated wth the expanson of steam durng ts flow. A decrease of 36 % n the densty of steam ndcates that the use of Bernoull s equaton for steam flow smulaton has certan lmtatons. Steam flow n ppelne s very fast. It takes about 35 s to travel 1000 m. In flud mechancs many emprcal relatons are used whch are based on the correlaton studes of expermental data. Therefore, the calbraton of a numercal model for the system to be studed s crucal. We performed the smulaton for a hypothetcal case. For a calbraton we need to consder real measurements. Presently, we are workng on the mplementaton of ths numercal approach for a geothermal ppelne network. It conssts of ncludng all the components lke valve, expanson-reducton, jont, etc. ACKNOWLEDGEMENTS Ths work was conducted under the project GeoSteamNet: a computer package for steam flow smulaton n a ppelne network, funded by our Insttute.
REFERENCES Bhave, P.R. and Gupta, R. (006), Analyss of Water Dstrbuton Networks Narosa Publshng House, Inda. Majumdar, A. (1999), Generalzed Flud System Smulaton Program (GFSSP) Verson 3.0, Report No. MG-99-90, NASA, USA. Patanker, S.V. (1980), Numercal Heat Transfer and Flud Flow, Hemsphere Publshng Corporaton, USA, 197p. Ruíz, A., Mendoza, A., Verma, M.P., García, A., Martínez, J.I. and Arellano, V. (010), Steam flow Balance n the Los Azufres Geothermal System, Mexco, Proc. World Geothermal Congress, n press. Smth, J.M. and Van Ness, H.C. (1975), Introducton to Chemcal Engneerng Thermodynamcs, 3 rd Ed., McGraw-Hll Kogakusha, Ltd., 63p. Verma, M. P. (003), Steam Tables for pure water as an ActveX component n Vsual Basc 6.0, Computer & Geoscences, 9, 1155-1163. Verma, M.P. (010), GeoSteamNet: 1. Thermodynamc data for steam flow n a ppelne network of geothermal system Ths volume.