Auto-connection technique in primitive part modelling
Contents
1. D 3 Ee C 42 C 43 C 44 C 4 5 v D 4 C 51 C 2 C 53 CSA C 5 5 D 5 C 1 C 5 5 A 1 5 5 A 2 5 5 A 3 5 5 0 A 11 1 A 24 1 C 2 C 5 1 A 1 5 1 A 2 5 1 A 3 5 1 0 A 11 2 0 Global overall matrix built by PP s C 3 C 1 5 A 1 1 5 A 2 1 5 A 3 1 5 0 A 11 3 0 C 4 2C 1 1 2A 1 1 1 A 2 1 1 A 3 1 1 2A 43 1 A 11 4 0 C 5 C 1 2 A 1 1 2 A 2 1 2 A 3 1 2 A 43 2 0 0 NE r s eua C 6 C 2 1 2A 1 2 1 A 2 2 1 A 3 2 1 2A 43 4040 C 3 CA C 5 0 0 82 c 13 C 7 C 2 2 A 1 2 2 A 2 2 2 A 3 2 2 A 43 5 0 0 0 C6 C7 C8 CLI x WM C 14 C 8 C 2 3 A 1 2 3 A 2 2 3 A 3 2 3 A 43 6 0 0 0 0 C 9 C 10 0 wi C 15 w C 9 C 3 2 A 1 3 2 A 2 3 2 A 3 3 2 A 43 8 0 0 Y i C 10 C 3 3 A 1 3 3 A 2 3 3 A 3 3 3 A 43 9 0 0 obal overall matrix C 11 C 2 4 A 1 2 4 A 2 2 4 A 3 2 4 A 43 11 0 built by conventional approach EE AG EE TER t C 12 D 5 B 1 5 B 2 5 B 3 5 0 B 11 1 B 24 1 C 13 D 1 B 1 1 B 2 1 B 3 1 2B 43 1 B 11 2 40 C 14 D2 B 1 2 B 2 2 B 3 2 B 43 2 0 0 C 15 D 3 B 1 3 B 2 3 B 3 3 B 43 3 0 0 Fig 4 Data flow in constructing insulated water pipe model from PP s 16 m 2 pu ed in m T a t pu 3 gt E 3 2 105 Tinerzec Fig 5 Outlet water temperature in response to pipe inlet disturbances 17 Table 1 List of primitive parts Thermal conduction solid to solid with ambient solid Surface convection with moist air with 22. View factor Working pressure Pa Heat injection rate W Dynamic viscosity kg m s Thermal conductivity W m K Prandtl number Entering air temperature C Working pressure Pa Heat injection rate W Dynamic viscosity kg m s Thermal conductivity W m K Prandtl number Entering fluid temperature C Working pressure Pa Dryness fraction Heat injection rate W Dynamic viscosity kg m s Thermal conductivity W m K Prandtl number Entering f luid temperature C
3. American Society of Heating Refrigerating and Air Conditioning Engineers Inc Atlanta 1975 3 Solar Energy Laboratory TRNSYS Version 15 user manual and documentation Madison Wisconsin Solar Energy Laboratory Mechanical Engineering Department University of Wisconsin 2000 4 ESRU ESP r a building and plant energy simulation environment user guide version 9 series University of Strathclyde Glasgow 2000 5 Crawley DB et al ENERGYPLUS New capabilities in a whole building energy simulation program Proceedings of Building Simulation 2001 the 7 International IBPSA Conference Aug 2001 Rio de Janeiro Brazil Vol 1 pp 51 58 6 Buhl W F Erdem FC Winkelmann FC Sowell EF Recent improvements in SPARK strong component decomposition multivalued objects and graphical interface Proceedings of Building Simulation 93 the 3 International IBPSA Conference Aug 16 18 1998 Adelaide Australia pp 283 289 7 Sahlin P Bring A Kolsaker K Future trends of the Neutral Model Format NMF Proceedings of Building Simulation 95 the 4 International IBPSA Conference Aug 14 16 1995 Madison Wisconsin USA pp 537 544 11 8 9 10 11 12 13 Sahlin P IDA modeler a man model interface for building simulation Proceedings of Building Simulation 93 the 3 International IBPSA Conference Aug 1993 Adelaide Australia pp 299 305 Clarke JA Energy Simulation in building design 2 ed Butt
4. In our auto connected PP system structure the node parameters control data and matrix coefficients can be grouped and exported as an input data file To facilitate data transfer transformation program is to be developed to convert such data file to a format recognized by other popular simulation software one example is the dck file for use in TRNSYS 3 Data flow 3 1 Data structure A key factor of developing a plant component as well as the entire HVAC system by auto connecting the PP s lies in how the parameter array can be formed with the data handled in the correct order Our discussions on the data structure will be based on a practical example of synthesizing an insulated water pipe model as illustrated in Fig 2 This insulated water pipe model is constructed by three PP s PP2 4 surface convection with ambient 1 node denoted by S PP1 1 thermal conduction for solid to solid 2 nodes denoted by M and N and PP4 3 flow upon surface for t phase fluid 3 nodes denoted by S WM and W1 Mathematical derivation of the conservation equation sets of this model has been reported elsewhere 13 and will not be repeated Instead the data format in the auto connected system will be introduced Three levels of basic data can be found in an auto connected PP system the node level the PP level and the component level A node is a region of physical space where the thermal state is described by the conservation e
5. Platforms i E I H AN NEN J Coefficient Generator l i a Solution O i Li mE Result Analysis I i l jocos Lef eed I Matrix Solver Process Result Files l 1 I SSS SS AE el eere enn ee geland a i Figures i 1 I 1 Li DNA Charts Satisfactory Outputs i i Contour Maps 1 Li Fig 1 Flow chart of PP modeling and simulation process 13 Node Name ldentifier Node type Material type Primitive Part Name in database Identifier Nodes list PP parameters Connection identifier Component Code Identifier Description PP list Fig 2 Data structure of PP auto connection 14 uonoeuuo epoN Uonoeuuo dd pipe model Fig 3 Insulated water pipe model formed from PP models 15 PP Sub matrices in PP library PP43 PP14 PP24 A431 A432 0 0 B 431 A 434 A 435 A 436 n res AA de dean AQ4n xIs BED A013 AQL0 N BaL 0 A 438 A 439 0 B 433 Global PP matrices Node order In defined problem 1 82 3 W ket 2 WM 4 W0 k 2 k 3 S S2 M S2 S 581 N S1 414 A 1 1 2 A LL3 AG EA 21 A L24 A L22 A L23 A 124 8 WM B 12 A 25 5 A 251 S1 BO A 3 55 MIS Br 3 12 A L31 A L32 A L33 A L34 WI B 13 A 2 5 A 244 1 2 B 22 er ij A 14 1 A L42 A 143 A 144 WO B 14 Global overall matrix C L1 C 12 C 13 C 14 C 15 T 52 D I lain C 22 C 23 C 24 a5 al lo C 31 C 32 C 33 C 34 C 3 5 x wi
6. There are two kinds of connection in the auto connected PP system the PP inner connection and the PP connection A PP inner connection describes the thermal and flow interaction between the nodes within one primitive part It is a program level connection Heat and mass transfer among the nodes is the physical behavior represented by the PP model This is described as a part of the PP attribute in PP models To exemplify this connection consider PP4 3 in Fig 3 that describes the thermal behavior of a single phase fluid like water when it flows along a solid surface The 3 fluid nodes denoted by WO WM and W1 are inner connected to model a fluid flow phenomenon This link is described as one feature of PP4 3 A connection between two PP s occurs whenthe user assigns a link between two PP s This is a user level connection which in essence means two nodes in different PPs physically merged to represent a region a single node of a plant component Numerically they represent one node in the solver and share the same information of this node In Fig 3 the solid node M of PP1 1 is connected and so combined with the solid node S of PP2 4 to epresent the thermal insulation layer of the water pipe M and S thus share the same technical data of the insulation layer Sometimes one of the two nodes may not occupy any physical space before merging such as the incoming node 3 3 2 Auto Connection Rules To guard against incorrect links be
7. general applications i e for the computation of dimensionless numbers or the heat transfer coefficients Listed in Table 2 are the common parameters associated with the nodes of the four different kinds Not all listed parameters are essential or even relevant for a specific application For example effective diameter is relevant for a solid node that represents an inner conduit surface but not a flat surface Encapsulated mass of air is insensitive to the simulation results in lots of moist airflow cases in practical air conditioning systems The list is prepared for preserving the generic nature of the PP approach and to help the users in specific cases Node type Incoming node this acts as input information and indicates the end of the former connection Contact node this associates with physical quantities for solving in the matrix solver Leaving node this also associates with physical quantities for solving in the matrix solver functionally it acts as output and indicates the start of another connection in the system Ambient node this provides the boundary condition data In our system the ambient node is a part of the PP sub models as inPP1 2 and PP2 4 Thus the time varying boundary conditions are read through the ambienttype PP sub models 3 2 2 PP Attributes PP data attributes include Name The name of a PP is referenced to the name of such PP sub model in the PP database For instance 011 is for PP 1 1 0
8. their physical properties features and operating conditionsshould be available for user input This technical data is node related rather than componentrelated The control data for use in the simulation process should also be stated at this stage The PP models so far developed are based on finite volume conservation equations The input and operating parameters are divided into three types state variables static data and dynamic data The state variables data i e nodal temperature and mass flow rates is updated through the solution process at every iteration step The static data including the control data is user defined and is read from either the user interface or the simulation input file The dynamic data such as fluid density and viscosity is computed based on the above two types of data as derived parameters With these three types of parameters all matrix coefficients can be obtained in every iteration step through the coefficient generator Their numerical values are passed to a matrix equation solver through which the solution at each time step is obtained The last step of the simulation procedure is result analysis Both graphical analysis and tabulated result files should be available User judgment and validation test can be exercised to observe the quality of the thermal model Sensitivity tests can be used to check the importance of certain energy flow path by either the addition or the deletion of some PP s in the model
9. 43 for PP4 3 and 101 for PP 10 1 lt Identifier This is the unique name of a PP in a specific problem It is automatically set each time when a PP is selected fromthe PP database Nodes list Nodes are the building blocks of PP sub models Thus the node list is an information table about the elements in a specific PP PP parameter An address table named PP parameter address database is built to store the numerical data These parameters include the thermal properties the physical attributes and the derived parameters The latter are determined through a communication interface with the user inputs Connection identifier This stores the link address of a specific PP in a problem which helps the program itself to locate such PP 3 2 3 Component Attributes Component attributes include Code The component code is based on the HVAC component classification It acts like the type and unit defined in TRNSYS or HVACSIM This attribute expands component utilization and its compatibility Identifier This is the unique name of a component in a specifi c problem It is automatically set when a specific problem is defined Description This is a text description used to outline a specific problem for future reference PP list PP s are the building blocks of a component PP list is an array providing information on these building elements of a specific component 33 PP connections 3 3 1 PP connection attributes
10. Auto connection technique in primitive part modelling T T Chow C L Song Division of Building Science amp Technology City University of Hong Kong Tat Chee Avenue Hong Kong China Corresponding author fax 852 27889716 email address bsttchow cityu edu hk Abstract In thermal simulation an approach to satisfy the diversifying simulation objectives is to provide a simulation environment that is capable to process the represented thermal system dynamically and to the desirable level of detail In the componentbased modeling approach adopted in contemporary building simulation software the components in the plant library often cannot truly reflect the actual system performance For some years efforts have been placed on the search for alternative simulation environment Primitive part PP modeling has been one of these efforts It possesses most of the attributes of a desirable environment such as flexible inputoutput independent hierarchical modeling structure detailed and unified fundamental approach This paper describes the background of PP modeling and the auto connection data flow structure that facilitates the model developer to synthesis simulation components from PP sub models through user interface Keywords Numerical simulation thermal models primitive part modeling data structure 1 Introduction Mathematical representations of thermal processes in building fabric and environmental control systems are highly nor li
11. a simulation network is the summation of the individual PP coefficients to generate the matrix coefficients of the global equation set 11 12 This super imposition rule makes the construction of plant components from PP s mathematically simple and elegant The advantages of the technique lie in the contribution towards component model standardization hierarchical model decomposition input output independent component representation controlled complexity of source code and interoperability of simulation platforms The technique provides a unified mathematical structure and has the potential to support extensibility through inheritance and offers a way to update models as the state of the art technology evolves A full implementation of the PP approach requires the availability of a graphical environment such that the PP synthesis of plant components can be worked through a userfriendly computer interface A description of the auto connection data flow structure is the main purpose of this article 2 PP Auto connection The flow chart in Figure 1 illustrates the PP modeling and simulation process through the user interface The user most often the model developer is expected to possess the technical know how about the nature and functions of each PP and the associated physical laws For a given simulation task the user should review and define well the purpose and the scope of the specific problem at hand He should develop the overall t
12. ability of the plant components whenever necessary as in component based modeling When facing a complex situ ation user could mix the PP synthesized component models with other empirical or semi empirical component models in the plant database Also feasible is to appropriately simplify the thermal system in an attempt to produce PP synthesized component models that operate with a reduced data set At this end PP modeling lends itself to adapt to various situations in treasuring for the state of the arts approach to numerical simulation A 4 row cooling coil can be represented in detail by an extensive 20 node PP synthesized model using PP4 3 PP4 4 and PP5 2 that can examine the condensation rate at each row or a simplified 3 node model using PP4 3 and PP10 3 with the heat removal from the moist air stream through a wetted surface in contact the heat extraction rate can be any arbitrary figure stated by the user These are where the flexibility lies 5 Conclusions Componentbased modeling approach in the numerical analysis of thermal system has been used for some years in the building industry Despite of the contemporary wide applications their limitations of restrictive and monolithic applications has led to efforts in searching for alternative approaches with targets at better code management promoting data sharing and satisfying everincreasing user aspirations Primitive part modeling is one of these efforts It possesses its uniquene
13. d D n represent the future time step and present time step coefficien amp in the global matrix respectively According to the super imposition rule the matrix coefficients C n m and D n can be given by the summation of PP coefficients i e Cn m y A i n m 3 i l D n Y Bin n i l To construct the insulated water pipe model the sub matrix templates of PP1 1 PP2 4 and PP4 3 in the PP library are respectively AQLI A 11 2 M Bu11 A 11 3 A 11 4 N B 11 2 A 24 x s B 24 1 6 S A 431 A 432 0 0 s B 43 A 434 A 435 A 436 A 43 11 x B 432 7 0 A 438 A 439 0 wo LB 493 Equation 8 gives the overall matrix equation template of this 4node component model denoted by S1 S2 WMand W1 with one incoming node denoted by WO as follows SI Q Ly C 12 8 0 0 D 1 C 24 C 22 C 2 3 0 0 D 2 5 x WM 8 3 2 C 33 C 3 4 C 35 i D 3 0 0 C 4 3 C 44 0 wa D 4 This overall matrix template for the pipe can be derived from the conventional approach by setting the conservation equations At the same time the coefficients of this matrix equation can be given by the summation of primitive part coefficients Figure 4 shows an example of the data flow process In this example the user made connections between nodes of different PP s in the order of PP4 3 then PP1 1 and then PP2 4 Accordingly the node order or the row number of the matrix from 1 to 5 is for S2 WM W1 WO an
14. d S1 This is purposefully made different from those in equation 8 to demonstrate that the data structure can support different assigned node order and can arrive at the same required values of the overall matrix coefficients If a specific simulation task is to investigate the thermal behavior of the insulated water pipe with respect to changes in inlet water temperature conditions equation 8 or the equivalent one built by PP s becomes the global matrix of the defined problem The program identifies the coefficients index of auto connected PP and dynamically set the matrix coefficients in the course of simulation Figure 5 gives the simulation results of two specific flow cases Case1 Step change in inlet water temperature from 5 C to 10 C Case2 Linear change in inlet water temperature from 5 C to 10 C in 10 seconds These results are exactly the same as the one developed directly from the conventional component based modeling approach 4 Flexibility Vs Complexity The benefit of applying PP s is to allow a flexible and generalized representation of the HVAC system under study A pre requisite for the convenient use of the PP s is then to make available in the input interface an adequate data set of commonly used parameters in system simulation These will include technical data such as physical dimensions like length thickness volume etc material properties like density specific heat conductivity etc and heat and mass t
15. election of system components from a plant component library and connecting them as if in the real situation This approachis adopted in contemporary simulation programs like TRNSYS 3 ESP r 4 and ENERGYPLUS 5 In real situation however the types and the topology of air conditioning system vary case by case Their accurate representation requires the simulation program be able to accommodate a large number of component types and levels of abstraction within each type Most of the component models in use at the present time are inflexible and based on range restricted empirical data With the continuing evolution in air conditioning technology there is a need to develop better component models and more flexible simulation environment Working in this direction were various endeavors the SPARK project 6 the Neutral Model Format NMF 7 and the IDA code 8 are some quoted examples From the viewpoint of thermal and fluid flow in HVAC equipment it has been found that some 27 sub model elements known as primitive parts or PP s as listed in Table 1 can be used to model the energy and mass transfer processes based on the finite volume conservation approach 9 10 The combination of these primitive parts could well represent different HVAC components at a range of levels of detail The process based nature of the PP approach differentiates it from the other approaches named above Mathematically the integration of PP s at each node of
16. erworth Heinemann 2001 Chow TT Clarke JA Dunn A Primitive parts an approach to air conditioning component modeling Energy and Buildings 1997 26 165 173 Chow TT Clarke JA Theoretical basis of primitive part modeling ASHRAE Transactions 1998 104 2 299 312 Chow TT Numerical modeling of fluid flow aid heat transfer processes by generic fundamental components Proceedings of HEFAT2002 the 1 International Conference on Heat Transfer Fluid Mechanics and Thermodynamics Vol 1 Part 1 Kruger Park South Africa April 2002 pp 302 307 Chow TT Generalization in plant component modelling Proceedings of Building Simulation 95 the 4th IBPSA International Conference Madison USA August 1995 pp 48 55 12 Function Description Simulation Task Define Problem Select PP s I Problem Description 1 1 1 1 1 1 Component Category Component Description Environment Details from PP Library 10PP Categories 27 Primitive Parts bod NE D 3 1 1 Connect FP s EE piu Fea alee GR Oet Connection Rules 1 1 1 1 l 1 1 7 1 Connection Index 1 1 1 l 1 1 1 Connection Validity amp gt ee PP Links Completeness LM Data Description H Imme I i Input Parameters l eet ELE i 1 Control Data amp Rules i EE i i Input Data Editing 1 Li aa i F Li P Transformation E Input Files for other SARA AA SEER AE E i 1 Simulation
17. hermal model and sort out the plant components in need In the process the user is to convert his physical plant into a simulation model making reference to the available PP s from the PP library The equipment should be categorized according to the HVAC common classification like fans and pumps flow conduits heat exchangers and so on This facilitates systemic categorization of plant components for future reference The available types of PP as at present allow the execution of dynamic thermal simulation on basic HVAC systems In its expanded application a modest extension of PP library is envisaged This means that a user should be able to develop his new PP sub models and inserts them intothe PP library when the need arises in future With the PP modeling approach the PP s are connected at the interface level with material type and fluid type identified A connection between two PP sub models is the merge of two selected nodes to become one which then represents the same spatial region within the overall thermal system The concept is different from the connection between two plant components in componentbased modeling approach The latter is virtually the pass of operation data from one component to another The builtin connection rules allow program checking of the validity of a connection say whether the connection is between two legal nodes at different PP s At the next step the technical data of the system component including
18. near and involving time dependent parameters Nevertheless heat transfer fundamentals are essentially the same applying to the building envelope and the plant system In thermal simulation what requires to satisfy the diversifying simulation objectives is a simulation environment that is capable to process the represented thermal system dynamically and to the required level of detail The way to achieve this is by establishing sets of conservation equations for different spatial regions and then arranging for solving these equations over the time span In a numerical sense this is to couple building and plant in the thermal model by gathering the energy and flow balance equations often in the matrix form for both the building side and the plant side The equation set is then solved either directly or by iteration 1 If we look back to the earlier stages of developments of building simulation in 70 s and 80 s the simulation software by that time provided detailed analysis of the thermal processes through building fabric but incorporated only with a simplified steady state plant energy performance representation 2 The oversimplification in system simulation resulted in the overwhelmingly use of arbitrary and predefined parameters This menu based approach was later on replaced by the component based simulation approach The component based approach allows the users to build a system network resembling the actual physical plant through the s
19. phase fluid with 1 phase fluid with ambient Surface radiation with local surface with ambient surface Flow upon surface for moist air 3 nodes for 2 phase fluid 3 nodes for 1 phase fluid 3 nodes for moist air 2 nodes for 1 phase fluid 2 nodes Flow divider and inducer Flow diverger for all fluid Flow multiplier for all fluid Flow inducer for all fluid Flow converger for moist air for 2 phase fluid for 1 phase fluid for leak in moist air from outside Flow upon water spray for moist air Fluid injection water steam to moist air Fluid accumulator 9 1 for moist air 9 2 for liquid 10 Heatinjection 10 1 to solid 10 2 to vapor generating fluid 10 3 to moist air Table 2 Parameters associated with different material types Basic Parameters Material Type Solid Moist air Single phase fluid Two phase fluid Type of material Total mass kg Contact surface area m Thermal resistance nf KW Heat injection rate W Mass flow rate kg s Encapsulated mass kg Moisture content kg kg dry air Density kg n Specific heat J kg K 2 Contact surface area m Type off luid Mass flow rate kg s Encapsulated mass kg Density kg n Specific heat J kg K Contact surface area n Type offluid Mass flow rate kg s Encapsulated mass kg Density kg n Latent heat of evaporation J kg Specific heats J kg K 2 Contact surface area m Length m Width m Effective diameter m
20. quation set So nodes are space oriented One level up is the PP s that are process oriented 27 of them are currently in the PP library Then the third level is the component level The plant components are function oriented and can be arranged to form an HVAC system model Besides these three levels of data there are information on the links between different nodes and different PP s These links are referred as the connection data each of which is characterized as either node connection or PP connection All kinds of data involved have distinctive attributes that differentiate them from the others 3 2 Data Attributes 3 2 1 Node Attributes Node attribute s include the following Name Identifier The name of a node is a unique identifier in a specific problem It is assigned automatically by the auto connected PP system when the user starts a new problem lt Material type Four material types or nature are being supported solid moist air single phase fluid and two phase fluid No restrictions are set for the physical shape orientation or dimensions of the nodes of different material types not until they have been selected and assigned a position in the simulation network For example a solid node can well describe the interior surface of a ventilating fan casing or the exposed surface to a duct heater all subject to the user definition For fluid nodes the technical data involved should be adequate for
21. ransfer coefficients at various flow conditions like piped flow wind induced airflow natural convection wet surfaces etc The input parameters should be readily available at a friendly and practical user interface For those less commonly used parameters the user is allowed to insert the additional information based on their own technical knowledge or experimental research of the situation under study Nevertheless a limitation of this generic approach is that the data in HVAC simulation is more complicated than in building simulation The data structure of the physical specification of the building structure composite walls transparent surfaces opened door or windows ground surfaces can be represented by a handful number of rules or laws In HVAC the available equipments are enormous and their working principles vary dramatically from one to another As the process related objects to obtain the overall model of the topologically complex thermal system is non trivial In the missing of an important heat transfer coefficient of a complex geometry for instance the user has to either dig out the most relevant formulae through rigorous literature search or do their own laboratory research to find out the correlation expression This is so far one basic barrier encountered in system simulation when opted for the explicit modeling approach From a practical point of view a step towards the wider use of PP s can be to narrow down the applic
22. ss in offering flexibility and uniformity through the use of generic process based elements The auto connected PP can provide a controlled complexity of the software architecture of an advancednumerical tool 10 A key factor in the data structure that makes possible the auto connection of PP s has been a systematic array of parameters and matrix coefficients that can be called upon efficiently and accurately The data attributes at each of the three levels node PP and component have been elaborated in the article Also explained are the connection attributes together with the connection theory and rules An insulated water pipe has been quoted as an example to demonstrate the details behind the rules and the techniques that allow the overall matrix coefficients be built dynamically A pre requisite for the use of the PP s is then to make available in the input interface an adequate data set of general parameters that are popularly used in system simulation Acknowledgement The authors express our thanks to the financial support of the Strategic Research Grant 7001114 from the City University of Hong Kong References 1 Clarke JA Prospects for truly integrated building performance simulation Proceedings of Building Simulation 99 the 6 International IBPSA Conference Sep 1999 Kyoto Japan Vol 3 pp 1147 1154 2 Stoecker WF Procedures for simulating the performance of components and systems for energy calculations 3rd ed
23. tween nodes from two different PP s auto connection rules are built in the code to help the user In our system several kinds of link errors and warning messages have been incorporated Incorrect links have been made an automatic failure These are to guard against the following possible human errors lt Nodes ofdifferent material types linked Connection could only occur between two nodes of the samematerial type For instance asolid node could only be connect ed with another solid node but not a flowing fluid node Incoming nodes linked Incoming node could only be linked with a leaving node or a contact node but not another incoming node V Leaving nodes linked a leaving node could only be linked with an incoming node or a contact node but not another leaving node 3 3 3 Connection theory In a given problem a typical PP matrix template with three nodes Jand L is in the following format A k i i Aki j Aid I Bi Alk ji A QG j j ACK j D x J BE j 1 A Li A Lj ALD L BD where k represents the k primitive part selected in the defined problem i j and represent numerically the node order of J and L in the overall matrix The global matrix of an overall system with S number of nodes can be represented by CALE L2 EE MY C Ls I D 1 C 2 1 2 C 2 5 II D 2 x 2 i C nl X C nm C n s D n C s l 2 C sm C s s S D s 2 In the equation C n m an
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