Connection equations with stream variables are generated in a model when using the # $ % () operator or the & ' %

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1 The two basic variable types in a connector potential (or across) variable and flow (or through) variable are not sufficient to describe in a numerically sound way the bi-directional flow of matter with convective transport of specic quantities such as specic enthalpy and chemical composition. The values of these specic quantities are determined from the upstream side of the flow i.e. they depend on the flow direction. When using across and through variables the corresponding models would include nonlinear systems of equations with Boolean unknowns for the flow directions and singularities around zero flow. Such equation systems cannot be solved reliably in general. The model formulations can be simplied when formulating two dferent balance equations for the two possible flow directions. This is not possible with across and through variables though. This fundamental problem is addressed in Modelica by introducing a third type of connector variable called stream variable declared with the prefix. A stream variable describes a quantity that is carried by a flow variable i.e. a purely convective transport phenomenon. The value of the stream variable is the specic property inside the component close to the boundary assuming that matter flows out of the component into the connection point. In other words it is the value the carried quantity would have the fluid was flowing out of the connector irrespective of the actual flow direction. The rationale of the definition and typical use cases are described in Appendix D. If at least one variable in a connector has the prefix the connector is called stream connector and the corresponding variable is called stream variable. The following definitions hold: The prefix can only be used in a connector declaration. A stream connector must have exactly one scalar variable with the prefix. [ ] For every outside connector [ ] one equation is generated for every variable with the prefix [!! " ]. For the exact definition see the end of section 5.. For inside connectors [ ] variables with the equations. prefix do not lead to connection Connection equations with stream variables are generated in a model when using the # $ % () operator or the & ' % () operator see sections 5. and 5.. [ ( ) * / 0 FluidPort Medium = Modelica.Media.Interfaces.PartialMedium; Medium.AbsolutePressure p "Pressure in connection point"; 8 Medium.MassFlowRate m_flow "> 0 flow into component"; / 0. : Medium.SpecicEnthalpy h_outflow "h close to port m_flow < 0"; / 0. : Medium.MassFraction X_outflow[Medium.nX] "X close to port m_flow < 0";. - ; FluidPort;

2 Modelica Language Specication. < = >? A B C D E ) F = " H I = F = H = J H = = "? = = K = = L M N A B C D E P Q =! = K = =! = L D R A S D T A B C D E P Q? U V ] In combination with the stream variables of a connector the # $ % W X operator is designed to describe in a numerically reliable way the bi-directional transport of specic quantities carried by a flow of matter. # $ % (v) is only allowed on stream variables v and is informally the value the stream variable has assuming that the flow is from the connection point into the component. This value is computed from the stream connection equations of the flow variables and of the stream variables. For the following definition it is assumed that N inside connectors m j.c (j=...n) and M outside connectors c k (k=...m) belonging to the same connection set [ ] are connected together and a stream variable h_outflow is associated with a flow variable m_flow in connector c / 0 FluidPort... 8 Real m_flow "Flow of matter; m_flow > 0 flow into component"; / 0. : Real h_outflow "Specic variable in component m_flow < 0". - ; FluidPort : ;. FluidSystem... FluidComponent m m... m N ; FluidPort c. Y Z / [ - c... c M ; / (m.c m.c); / (m.c m.c); / (m.c m N.c); / (m.c c ); / (m.c c ); / (m.c c M ); ; FluidSystem; ~ ƒ ~ ~ ƒ ˆ ˆ ˆ ˆ ˆ ˆ \ ] ^ _ ` a b c d b e f g h i j k h l m n k o p q r m s t u i v p t w x y z h { q y }

3 Æ Æ» û [ Š Œ Ž Ž Œ Ž Ž Ž Ž Ž Œ Ž š œ Š Œ Š Š Œ Ž Ž Œ Ž š Ž š Ž Š Œ Ž Ž š Ž Ž ž Ž Ÿ ] With these prerequisites the semantics of the expression (m i.c.h_outflow)is given implicitly by defining an additional variable h_mix_in i and by adding to the model the conservation equations for mass and energy corresponding to the infinitesimally small volume spanning the connection set. The connect equation for the flow variables has already been added to the system according to the connection semantics of flow variables defined in section 9.. // Standard connection equation for flow variables 0 = sum(m j.c.m_flow ª «j :N) + sum(-c k.m_flow ª «k :M); Whenever the operator is applied to a stream variable of an inside connector the balance equation of the transported property must be added under the assumption of flow going into the connector // Implicit definition of the ± «² ³ µ operator applied to inside connector i 0 = sum(m j.c.m_ flow*( m j.c.m_flow > 0 ª «j==i ± ² h_mix_in i ² ¹ ² m j.c.h_outflow) ª «j :N) + sum(-c k.m_flow* ( c k.m_flow > 0 ± ² h_mix_in i ² ¹ ² ± «² ³ (c k.h_outflow) ª «k :M); ± «² ³ (m i.c.h_outflow) = h_mix_in i ; 75 Note that the result of the (m i.c.h_outflow) operator is dferent for each port i because the assumption of flow entering the port is dferent for each of them. º»»» à ¼ ½ ¾ ¼ À ¼ Á  ¼ Ä À Å Á ½ À ¼ ¼ À ¼ ¼ À. // Additional connection equations for outside connectors ª «q :M ª ª Ç 0 = sum(m j.c.m_flow*( m j.c.m_flow > 0 ± ² h_mix_out q ² ¹ ² m.c.h_outflow) ª «j :N) + j sum(-c.m_flow* ( c.m_flow > 0 ª «k==q ± ² k k h_mix_out q ² ¹ ² ± «² ³ (c k.h_outflow) ª «k :M); c.h_outflow = h_mix_out q ; q ² ª «; Neglecting zero flow conditions the solution of the above-defined stream connection equations for instream values of inside connectors and outflow stream variables of outside connectors is (for a derivation see Appendix D): ± «² ³ (mi.c. h_outflow) := (¹ ( ³ Ê (-m j.c.m_flow0)* m j.c.h_ou tflow ª «j ca t( :i- i+:n) + ¹ ( ³ Ê ( c k.m_flow 0)* ± «² ³ (c k.h_outflow) ª «k :M))/ (¹ ( ³ Ê (-m j.c.m_flow0) ª «j cat(:i- i+:n) + ¹ ( ³ Ê ( c k.m_flow 0) ª «k :M)); // Addit ional equations to be generated for outside connectors q ª «q :M ª ª Ç c q.h_outflow := ² ª «; (¹ ( ³ Ê (-m j.c.m_flow0)* m.c.h_outflow ª «j :N) + j ¹ ( ³ Ê (c k.m_flow0)* ± «² ³ (c k.h_outflow) ª «k cat(:q- q+:m))/ (¹ ( ³ Ê (-m j.c.m_flow0) ª «j :N) + ¹ ( ³ Ê ( c k.m_flow 0) ª «k cat(:q- q+:m))); Ë Š Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö Ø Ù Ú Û Ü Ý Þ Ú ß à Œ š ž Š Œ Ž Ž Œ Ž Š Ž Ž Š Œ Š Œ á â Š Š Ÿ Š Õ Ö Ø Ù Ú Û Ü Ý Þ Ú ß Ž Ž Ÿ Ž š ž Œ Ž Ž Œ á š Ž Ž ž Œ Ž Ž Œ Ž ž Š Ž Š š á Ž ž Œ Ž Ž Œ Œ Ž á š ž Œ š

4 ô ò»»» 76 Modelica Language Specication. If the argument of () is an array the implicit equation system holds elementwise i.e. () is vectorizable. The stream connection equations have singularities and/or multiple solutions one or more of the flow variables become zero. When all the flows are zero a singularity is always present so it is necessary to approximate the ä š Š å Õ å ç Ý Þ Ú ß è Õ Ö å ç Ý Þ Ú ß è é solution in an open neighbourhood of that point. [ã Š Ž š Ž á ž Ž Œ š ž Ž ž Ì Í Î Ï Ð Ñ Ò Ó Ô à Œ Ž Š Ž š ]. It is required that the () operator is appropriately approximated in that case and the approximation must fulfill the following requirements: š š Ž š Œ Œ Œ š Œ. (m i.c.h_outflow) and (c k.h_outflow) must be unique with respect to all values of the flow and stream variables in the connection set and must have a continuous dependency on. them. Every solution of the implicit equation system above must fulfill the equation system identically [ ] provided the absolute value of every flow variable in the connection set is greater as a small value ( m.c.m_flow > eps m.c.m_flow > eps... c M.m_flow > eps). [ ê ž Ž Š á š Ž Š œ Ž Ÿ Ž Ž Œ Ž ž ž ë ì í î ï í ñ ò ± «² ³ (m.c.h_outflow) = m.c.h_outflow; ì í ó ï í ñ ò ± «² ³ (m.c.h_outflow) = m.c.h_outflow; ± «² ³ (m.c.h_outflow) = m.c.h_outflow; ì í î ï í î ò ± «² ³ (m.c.h_outflow) = ± «² ³ (c.h_outflow); // Additional equation to be generated c.h_outflow = m.c.h_outflow; ª ± ² «õ ³ ¹ ² ¹ m j.c.m_flow.min >= 0 ª «³ j = :N with j <> i ³ ck.m_flow. <= 0 ª «³ k = :M ± ² ± «² ³ (m i.c.h_outflow) = m i.c.h_outflow; ² ¹ ² si = ¹ ( ³ Ê (-m j.c.m_flow0) ª «j cat(:i- i+:n) + ¹ ( ³ Ê ( c k.m_flow 0) ª «k :M); ± «² ³ (m i.c.h_outflow) = (¹ (Ç ª ¹ ± ö ² ³ Ê (-m j.c.m_flows i)* m j.c.h_outflow) + ¹ (Ç ª ¹ ± ö ² ³ Ê (c k.m_flows i) * ± «² ³ (c k.h_outflow)))/ (¹ (Ç ª ¹ ± ö ² ³ Ê (-m j.c.m_flows i)) + ¹ (Ç ª ¹ ± ö ² ³ Ê (c k.m_flows i ))) ª «j in :N ³ i <> j ³ m j.c.m_flow.min < 0 ª «k in :M ³ c k.m_flow. > 0 // Additional equations to be generated ª «q :M ª ª Ç m j.c.m_flow.min >= 0 ª «³ j = :N ³ c k.m_flow. <= 0 ª «³ k = :M ³ k <> q ± ² c q.h_outflow = 0; ² ¹ ² í s q (¹ ( ³ Ê (-m j.c.m_flow0) ª «j :N) + ¹ ( ³ Ê ( c k.m_flow 0) ª «k cat(:q- q+:m))); c q.h_outflow = (¹ ( Ç ª ¹ ± ö ² ³ Ê (-m j.c.m_flows q)* m j.c.h_outflow) + ¹ (Ç ª ¹ ± ö ² ³ Ê (c k.m_flows q) * ± «² ³ (c k.h_outflow)))/ (¹ (Ç ª ¹ ± ö ² ³ Ê (-m j.c.m_flows q )) + ¹ (Ç ª ¹ ± ö ² ³ Ê (c k.m_flows q )))

5 77 ² ª «; Š Ç ª ¹ ± ö ² ª «j :N ³ m j.c.m_flow.min < 0 ª «k :M ³ k <> q ³ c k.m_flow. > 0 ³ Ê (-m j.c.m_flows i ) Š š ž á š Œ Š Š ë ä ø ù ú Œ Ù œ û ü ý ù ú Œ Ù Ù œ ù ú Œ Ù œ þ ÿ ú þ ý œ Š ÿ ú œ Œ ž Œ á ù Œ Œ š ž Ž Ž Š š ž á þ þ œ Š œ Œ ž Œ á ù Œ Œ š Ž Ž ä Ÿ š Ž Œ Ž Ž š Ž ø ü ë Ç ª ¹ ± ö ² ³ Ê (-m j.c.m_flow s i ) = ³ Ê (-m j.c.m_flow eps); // so s i is not needed Š Œ ž Ž Ž œ Š Š ä Ž ž Ž œ Š Ž œ ë // Define a "small number" eps ( ª ³ (v) is the nominal value of v) eps := relativetolerance* ( ª ³ (m j.c.m_flow)); // Define a smooth curve such that alpha(s i >=eps)= and alpha(s i <=0)=0 alpha := ¹ ª ª ± ( s i > eps ± ² ² ¹ ² s i > 0 ± ² (s i /eps)^*(-*(s i /eps)) ² ¹ ² 0); // Define function Ç ª ¹ ± ö ² ³ Ê (vs i ) as a linear combination of ³ Ê (v0) // and of eps along alpha Ç ª ¹ ± ö ² ³ Ê ((-m j.c.m_flows i ) := alpha* ³ Ê (-m j.c.m_flow0) + (-alpha)*eps; Š ž Ž Š Ž Ž ž Œ š ž Ž â Ž ž ä Ë Š Ž Š Œ Ë ý ÿ ý ø š Ž Œ Ž Ž Œ ž Š Œ Œ Ž ž Ž Ÿ š Ÿ Ÿ ž ù š Ž Ÿ ü Ž ž Ë ý ý ø Ž ù ù Ž Œ Ž Ž Œ Ž ü Š š Ì Í Î Ï Ð Ñ Ò Ó Ž ž Ž Ž Ž ù Ž š Ž Ž Š ž ž Š Œ Š Š Ÿ Ž ž Ž Ž š Ž Ž Ž Ž Š œ Ž Ÿ Š ž Š Ž Ž ë! " # $ % & % ' _( ) * + ( -. # $ % & % # _+ ( -. & / % ' _( ) * + ( -. & / % # _+ ( - 0! : ; : < = 7 A # $ % & % # B + ( -. C D E E F H I C J F K " L L M N " P " Q " P Q C ) # : ; : < = 7 A # $ % & % # B + ( -. C D E E F H I C E R L S T U V W X Y Z 7 # D % & % ' _( ) * + ( - E # $ % & % ' B ( ) * + ( - [ 9 \ ] ^ F _? ` a # ] % & % # B + ( - % # _ b ^ K E c d I " Q! " d I! # $ % & % ' _( ) * + ( - " " ] ^ F _ e b f # $ % & % # _+ ( - % # _ b ^ K N " Q P Q " " "! Q L Q! Q " Q " Q Q N "!! N " " "! "!! " Q 0 Q g " Q Q g " " L "! Q h Q i L j k L " " " Q Q h Q i L Q l j k N " "!! L " " Q ] The m n o p m q r o s t m u (v) operator is provided for convenience in order to return the actual value of the stream variable depending on the actual flow direction. The only argument of this built-in operator needs to be a

6 78 Modelica Language Specication. reference to a stream variable. The operator is vectorizable in the case of vector arguments. For the following definition it is assumed that an (inside or outside) connector c contains a stream variable h_outflow which is associated with a flow variable m_flow in the same connector c: v w x y v z { x } v ~ (port.h_outflow) = port.m_flow > 0 x } } z ƒ } { x } v ~ (port.h_outflow) port.h_outflow; [N " Y V Y U V W X Y Z! L P der(u) = c.m_flow*actualstream(c.h_outflow); // ()energy balance equation h_port = actualstream(port.h); // ()monitoring the enthalpy at a port R " ˆ " " " Q " L " " L! Q L " " L N " Q " " " P Q " k Q L " "! R L!! " Š " Q " " M " ˆ L " "! g Œ Q " Q L " " Ž " " L " " R " " L L " " L! " P! " R " Ž " " " Š " " " " M " Œ L "! " Q L " Q " "! Q Q Q Q M Q " " N " Q " Q " "! L " Q " L! g "! " Q! " ]

7 ³ h ž ª h Ÿ Ÿ ª ª ž ± ª ª 7 This appendix contains a derivation of the equation for stream connectors from Chapter 5. Consider a connection set with connectors. The mixing enthalpy is defined by the mass balance 0... and the energy balance 0... with œ ž š 0 0 Herein mass flow rates are positive when entering models (exiting the connection set). The specic enthalpy represents the specic enthalpy inside the component close to the connector for the case of outflow. Expressed with variables used in the balance equations we arrive at: ² ± ««± arbitrary 0 0 While these equations are suitable for device-oriented modeling the straightforward usage of this definition leads to models with discontinuous residual equations which violates the prerequisites of several solvers for nonlinear equation systems. This is the reason why the actual mixing enthalpy is not modelled directly in the model equations. The stream connectors provide a suitable alternative.

8 µ µ µ µ µ µ ³ 8 Modelica Language Specication. Figure D-. Exemplary connection set with three connected components and a common mixing For simplicity the derivation of the instream() operator is shown at hand of model components that are connected together. The case for N connections follows correspondingly. The energy and mass balance equations for the connection set for components are (see above): 0 ¼ ½ ¾ ¹ º» ¼ ½ ¾ ¹ º» ¼ ½ ¾ ¹ º» (b) The balance equations are implemented using a () operator in place of the piecewise expressions taking care of the dferent flow directions: Å Æ Ç 0 0 Å Æ Ç 0 0 Å Æ Ç Equation (a) is solved for Å Æ Ç Ê Ë Ì Using (b) the denominator can be changed to: (a) (a) (b)

9 Ô ç Æ ç Æ Ô Í Ô Ô Í Í Å Æ Ç Above it was shown that an equation of this type does not yield properly formulated model equations. In the streams concept we therefore decide to split the energy balance which consists of dferent branches depending on the mass flow direction. Consequently separate energy balances are the result; each valid for specic flow directions. In a model governing equations have to establish the specic enthalpy Î Ï ÐÑ Ò Î Ó of fluid leaving the model based on the specic enthalpy of fluid flowing into it. Whenever the mixing enthalpy is in a model it is therefore the mixing enthalpy under the assumption of fluid flowing into said model. Ø Ù Ú Û Ü Ý Þ We establish this quantity using a dedicated operator Î Ï ÐÑ Ò Î Ó Õ Í Ö 0. This leads to three dferent incarnations of Õ Í Ö 9 (n in the general case). This is illustrated in the figure below. For the present example of three components in a connection set this means the following. ß à á â ã ä å ß à á â ã ä å ß à á â ã ä å è é ê ë ì í î ï ñ ò ó í ô õ ö ì ø ù ú û û í ù ü é ú û ý í ü þ é ü ÿ ü ÿ ì í í ù ú û û í ù ü í ù ú ô õ ú û í û ü ý In the general case of a connection set with components similar considerations lead to the following. ß à á â ã ä å... ;... ; 0 0

10 8 Q! P!!! Q P P P Q Q Q P 0 Modelica Language Specication. For this case the return value of the instream() operator is arbitrary. Therefore it is set to the outflow value. " # $ % & ' (! " # $ % & ' (! In this case instream(.) is continuous (contrary to ) * + ) and does not depend on flow rates. The latter result means that this transformation may remove nonlinear systems of equations which requires that either simplications of the form a*b/a = b must be provided or that this case is directly treated. -. / - / 0 0 This case occurs when a one-port sensor (like a temperature sensor) is connected to two connected components. For the sensor the min attribute of the mass flow rate has to be set to zero (no fluid exiting the component via this connector). The suggested implementation results in the following equations: 9 : ; < = H I J K L M N A B CD E A F A B CD E A F R S T U V W X P P 0 _ ` a b c d e f g h i j k l m n d o d c ` d o p q r r d p s ` q r q t l b n s ` m n d l q u d n o v ` s w o s c d k l p q r r d p s q c o For the two components with finite mass flow rates (not the sensor) the properties discussed for two connected components still hold. The connection set equations reflect that the sensor does not any influence by discarding the flow rate of the latter. In several cases a non-linear equation system is removed by this transformation. However instream(..) results in a discontinous equation for the sensor which is consistent with modeling the

11 convective phenomena only. The discontinuous equation is uncritical the sensor variable is not used in a feedback loop with direct feedthrough since the discontinuous equation is then not part of an algebraic loop. therwise it is advisable to regularize or filter the sensor signal. x -. - / - y z / y / { } } ~ / / - If uni-directional flow is present and an ideal splitter is modelled the required flow direction should be defined in the connector instance with the min attribute (the attribute could be also defined however it does not lead to simplications): ƒ m Fluidport c(m_flow( =0));... m; Consider the case of 0 and all other mass flow rates positive (with the min attribute set accordingly). Connecting m.c with m.c and m.c such that m.c.m flow.min = 0; // (-m.c.m_flow0) = 0 m.c.m_flow.min = 0; // (-m.c.m_flow0) = 0 results in the following equation: H I J K L M N The instream() operator cannot be evaluated for a connector on which the mass flow rate has to be negative by definition. The reason is that the value is arbitrary which is why it is defined as follows. H I J K L M N A B CD E A F : A B CD E A F For the remaining connectors the instream() operator reduces to a simple result. 0 0 H I J K L M N H I J K L M N 0 0 Again the previous non-linear algebraic system of equations is removed. This means that utilizing the information about uni-directional flow is very important. To summarize all mass flow rates are zero the balance equations for stream variables () and for flows () are identically fulfilled. In such a case any value of h_mix fulfills () i.e. a unique mathematical solution does not exist. This specication only requires that a solution fulfills the balance equations. Additionally a recommendation is given to compute all unknowns in a unique way by providing an explicit formula for the inflow(..) operator. Due to the definition that only flows enter this formula where the corresponding min attribute is neither zero nor positive a meaningful physcial result is always obtained even in case of zero mass flow rate. As a side effect non-linear equation systems are automatically removed in special cases like sensors or uni-directional flow without any symbolic transformations (no equation must be analyzed; only the min - attributes of the corresponding flow variables).

12 Modelica Language Specication.

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