General Flow Equation

steady flow of gas in a pipe is derived from Bernoulli's equation

since pressure decreases along the pipe in the direction of flow
- because of the loss of pressure head to overcome friction
the density of the gas also decreases

for steady flow the mass flow is constant along the pipe

Given $A$, the pipe cross-sectional area, is constant then we have:

$\rho_1 w_1 = \rho_2 w_2$

thus as the density decreases the velocity increases
- and the kinectic energy of the gas must increase

derivation of the general flow equation involves a number of simplifying assumptions:

steady flow
- isothermal flow due to heat transfer with the surroundings through the pipe wall
  - negligible kinectic energy change in the pipe
    - constant compressibility of the gas over the length of the pipe
      - validity of the Darcy friction loss relationship across the pipe
        
        constant friction coefficient along the pipe length

CONUS Gas Flow

Pipeline Modeling

Complete gas flow mathematical model

BlendPATH
- network formulation
  - system of nonlinear equations
    - applies mass balance for every node $L_i + W_{fuel,i} = \sum\limits_{p=1}^{N_{Pipes}} a_{i,p}m_p \forall i \in N_{Nodes}$
      - where $m_p$ is the mass flow rate through pipe p
      - $a_{i,p}$ is a branch-node incident value of 1,0 or -1
        
        denoting the connection of pipe $p$ to node $i$
        
        with 0 denotes no connections
        
        and 1 or -1 denotes flow into or out of the node, respctively
      - the summation of mass flow rates representing external supplies and demands at node $i$ is denoted as $L_i$
      - fuel consumption at node $i$ is represented as $W_{fuel,i}$ if the node represents a compressor station inlet $\mathbf{Am} = \mathbf{L} + \mathbf{W_{fuel}}$
        
        where $P_i$ is the square of pressure at node $i$
        
        and $\Delta P_p$ is the difference of the squared nodal pressures through pipe $p$ $-A^TP=ΔP
        
        where $\mathbf{P}$ is the vector of the squared nodal pressurse in dimension $N_{Nodes}$
        
        and $\Delta \mathbf{P}$ is the vector form of $\Delta P_p$ in dimension $N_{Pipes}$
        
        $A_T$ is the transpose of the branch-nodal incidence matrix
        
        $m = \Phi(\Delta P)$
        
        where $\Phi(\Delta P)$ is the vector functions relating the pipe pressure drop to the mass flow rate
        
        substitue it in for $\Delta P$ $m=\Phi(-\mathbf{A^T P})$
        
        substitute above equation to relate nodal pressures to external mass flow rates at nodes representing supplies or demands $\mathbf{A}[\Phi(-A^T P)]= L+W_{fuel}$
        
        could also be rearranged into a vector of nodal error functions $\mathbf{F(P)}$ of dimension $N_{Nodes}$ as: $\mathbf{F(P) = [\Phi(-A^T P)]- L - W_{fuel}}$
        
        where $\mathbf{F(P^*)=0}$ when simulation solution method determines the vector of nodal pressure $\mathbf{P}^*$ that satisfies the nodes representing supplies and demands
        
        can be expressed algebraicly: $f(P_i) = -L_i - W_{fuel,i} + \sum\limits_{j=1}^{N_{Nodes}} a_{i,j} (A(\frac{M_w}{ZRT}\frac{D}{fL})^{0.5}\sqrt{a_{i,j}(P_i - P_j)}) \forall i \in N_{Nodes}$
        
        where $f(P_i)$ is the element-wise expression for the vector of nodal error functions
        
        and $j$ is an index representing an adjacent node connected by pipe $p$ to node $i$
        
        pipe $p$ represents a branch connecting two nodes, $i$ and $j$
        
        the pipe index, $p$, is substituted with $j$, another node index of dimension $N_{Nodes}$
        
        to represent adjacent node connected by pipe $p$ $p \Leftrightarrow (i,j)$ and $(i,j) \in N_{Pipes}$
        
        within this nodal model representation of pipeline network
GasModels
- network formulation
  - nonconvex models of all gas network components junctions
    - Each junction $i \in \mathcal{N}$ in the (natural gas transmission) network is associated with a pressure variable, $p_i$
      - Operational limits require that this pressure ranges between predefined
        
        lower and upper bounds, denoted by $\underline{p_i}$ and $\bar{p_i}$, respectively i.e. $0 \leq \underline{p_i} \leq p_i \leq \bar{p_i}, \forall_i \in \mathcal{N}$
      pipes
    - Each pipe satisfies the Weymouth Equation
      - which relates the mass flow rate $f_{ij}$ through the cross-sectional area of the pipe
        
        to the pressure $p_i$ and $p_j$ at the two end-points of the pipe $p^2_i -p^2_j = w_{ij}f_{ij}|f_{ij}|, \forall(i,j) \in \mathcal{P}$
        
        where the mass flow resistance $w_{ij}$ is related to $\gamma_{ij} := (16\gamma_{ij})/(\pi^2 D^4_{ij})$
      compressors
    - Each compressor $(i,j) \in \mathcal{C}$ boosts the pressure at the down-stream junction $j \in \mathcal{N}$ by a variable scalar $\alpha_{ij}$
      - and is assumed to have negligible length
      - bidirectional compressors do not exist
        
        although each compressor may or may not allow for uncompressed flow in the reverse direction
        
        i.e., from $j$ to $i$
      - In GasModels study these distinct behaviors of compressors are captured by three conditional sets of constraints
        
        Where uncompressed reverse flow is prohibited $\underline{\alpha_{ij}} \leq p_j \leq \bar{\alpha}_{ij}p_i, \forall(i,j) \in \mathcal{C} : \underline{f_{ij}} \geq 0$
        
        where $\underline{\alpha_{ij}}$ and $\bar{\alpha_{ij}} \in \mathcal{C}$ are minimum and maximum compressor ratios
        
        Where reverse flow is allowed and the minimum compression ratio is equal to one $\underline{\alpha_{ij}} \leq p_j \leq \bar{\alpha}_{ij}p_i, \forall(i,j) \in \mathcal{C} : \underline{f_{ij}} < 0 \wedge \underline{\alpha_{ij}} = 1$ $f_{ij}(p_i - p_j) \leq 0, \forall(i,j)\in \mathcal{C} : \underline{f}_{ij} < 0 \wedge \underline{\alpha}_{ij} = 1$
        
        if $f_{ij} < 0$ (i.e., where there exists reverse flow)
        
        then $p_i = p_j$ (i.e., there is no change in pressure across the compressor)
        
        Where uncompressed reverse flow is allowed and the minimum compression ratio is not equal to one
        
        the behavior must be modeled disjunctively
        
        for each compressor, a binary variable $y_{ij} \in {0,1}$ is introduced to denote the direction of flow through the compressor
        
        where $y_{ij}=1$ implies flow from $i$ to $j$ and $y_{ij}=0$ implies flow from $j$ to $i$
        
        the pressure at the junctions that connect each compressor are then modeled as: a. a$y_{ij} \in {0,1}, \forall(i,j) \in \mathcal{C} : \underline{f}_{ij} < 0 \wedge \underline{\alpha}_{ij} \neq 1$ b. $p_j \leq \bar{\alpha}_{ij}p_i + (1-y_{ij})\bar{p}_j, \forall(i,j) \in \mathcal{C} : \underline{f}_{ij} < 0 \wedge \underline{\alpha}_{ij} \neq 1$ c. $\underline{\alpha}_{ij}p_i \leq p_j + (1-y_{ij})\bar{p}_i, \forall(i,j) \in \mathcal{C} : \underline{f}_{ij} < 0 \wedge \underline{\alpha}_{ij} \neq 1$ d. $p_i - p_j \leq y_{ij}\bar{p}_i, \forall(i,j) \in \mathcal{C} : \underline{f}_{ij} < 0 \wedge \underline{\alpha}_{ij} \neq 1$ e. $p_j - p_i \leq y_{ij}\bar{p}_i, \forall(i,j) \in \mathcal{C} : \underline{f}_{ij} < 0 \wedge \underline{\alpha}_{ij} \neq 1$
        
        b. and c. ensure that when a compressor's flow is positively directed
        
        the pressure at node $j$ is modeled according to the compression ratio bounds
        
        d. and e. ensure that when flow is reversed
        
        the pressure on both sides of the compressor are equal
        
        valves
      - Each valve is modeled as a resistor
        
        the operating status of each valve $(i,j) \in \mathcal{V}$ is indicated using a binary variable $z_{ij} \in {0,1}$
        
        where $z_{ij}=1$ corresponds to an open valve and $z_{ij}=0$ to a closed valve
        
        These variables constrain the mass flow across each valve as: $\underline{f}_{ij}z_{ij} \leq f_{ij} \leq \bar{f}_{ij}z_{ij}, z_{ij} \in \{0,1\}, \forall(i,j) \in \mathcal{V}$
        
        when a valve is open, the pressures at the junctions connected by that valve are equal
        
        when a valve is closed, these pressures are decoupled
        
        this phenomenom is modeld via the following set of disjunctive constraints: $p_i \leq p_j + (1-z_{ij})\bar{p}_i, \forall(i,j) \in \mathcal{V}$ $p_j \leq p_i + (1-z_{ij})\bar{p}_j, \forall(i,j) \in \mathcal{V}$
        
        regulators
      - The operating status of a regulator is given using a binary variable $z_{ij} \in \{0,1\}$
        
        where $z_{ij} = 1$ and $z_{ij} = 0$ indicate active and inactive statuses
        
        mass flow across each regulator is then governed by: $f_ijz_ij ≤ f_ij ≤ |{f}_ij ≤ |{f}_ijz_ij, z_ij ∈ $
      - Each regular $(i,j)$ is associated with multiplicative scaling factor, $\alpha_{ij}$
        
        that defines the relationship between $p_i$ and $p_j$ when the regulator is active, i.e. $\alpha_{ij}p_i = p_j$
        
        this factor is constrained by operating limits as $\underline{\alpha}_{ij} = 0 \leq \alpha_{ij} \leq \bar{\alpha}_{ij} = 1$
        
        as for valves , the pressures at the junctions connected by a regulator are decoupled when the regulator is inactive
        
        modeled by: a. $f_{ij}(p_i - p_j) \geq 0, \forall(i,j) \in \mathcal{W}$ b. $\underline{\alpha}_{ij}p_i \leq p_j + (1-z_{ij})\underline{\alpha}_{ij}\bar{p}_i, \forall(i,j) \in \mathcal{W}$ c. $p_j \leq \bar{\alpha}_{ij}p_i + (1-z_{ij})\bar{p}_j, \forall(i,j) \in \mathcal{W}$
        
        a. ensures that each mass flow is in the same direction as the pressure loss
        
        b. and c. ensure the pressure at node $j$ resides within the scaled bounds of $p_i$ when the control valve is open
- maximal load delivery formulation
  - which seeks to determine a feasible operating point for a damaged gas pipeline network that maximizes the delivery of prioritized load
  three variants of the problem are presented in order of successive reformulations and relaxations that, though they increase problem complexity, decrease nonconvexity and hence computational difficulty
  - extend the nonconvex feasiblity constraints of MINCP-F to formulate the initial MLD problem
  - introduce an exact mixed-integer nonconvex quadratic reformulation of this MLD problem
  - introduce a mixed-integer convex quadratic relaxation reformualation of the MLD problem
  also present mixed-integer quadratic reformulations for a broader set of gas pipeline network components and introduce convex relaxation techniques for components that were not explicitly considered
  - mixed-integer nonconvex formulation
    - a damaged gas network requires the exclusion of components from the model
      - implies an imbalance in satisfying all demands of the original system
      - the set of excluded components comprises both damaged and connected components
        
        e.g. a damaged junction $i \in \mathcal{N}$ implies nonoperational status of all node-connecting components $\delta^+_i \cup \delta^-_i$ tilde-based notation to define set of components that are still functional
        
        e.g. $\tilde{P} \subseteq P$ denotes the set of still operational pipes
    - receipts and deliveries are treated as dispatchable
      - implying that the previous constant supplies and demands, $\bar{s}_k, k \in \mathcal{R}_i$, and $\bar{d}_k$, $k \in \mathcal{D}_i$ for $i \in \mathcal{N}$ becomes variable bounded as: a. $0 \leq s_k \leq \bar{s}_k, \forall k \in \mathcal{R}_i, \forall i \in \tilde{\mathcal{N}}$ b. $0 \leq d_k \leq \bar{d}_k, \forall k \in \mathcal{D}_i, \forall i \in \tilde{\mathcal{N}}$
        
        where $\bar{s}_k$ and $\bar{d}_k$ denote original supply and demands
        
        under variable loads, previous mass conservation become
        
        $\sum\limits_{(i,j)\in \tilde{\delta}^+_i}} f_{ij} - \sum\limits_{(j,i)\in \tilde{\delta}^-_i} = \sum\limits_{k\in \mathcal{R}_i}s_k - \sum\limits_{k\in \mathcal{D}_i}d_k, \forall i \in \tilde{\mathcal{N}}$
        
        parameters $\beta_k \geq 0, k \in \mathcal{D}$ are introduced to denote load restoration priorities, if none available then values of one can be used instead
        
        Maximization of prioritized load delivered then implies the objective function: $\eta(d) = \sum\limits_{i \in \tilde{\mathcal{N}}} \sum\limits_{k\in\mathcal{D}_i} \beta_k d_k$
    The mixed-integer nonconvex , nonlinear MLD problem formulation is then
    - maximize objective function $\eta(d)$
    - subject to supply and demand bounds, conservation of mass flow, and supplies and withdrawals of gas are not constrained to the nodal balance and mass conservation
Pandapipes
- mathematical model formulation
  - for any piping grid, the laws of Kirchoff apply
    - Kirchoff's junction rule
      - for any node/junction in a physical system or circuit
        
        the sum of current flowing into that node/junction is equal to the sum of currents flowing out of that node $\sum\limits_{i=1}^n I_i = 0$
        
        where $n$ is total number of branches with currents flowing towards or away from the node
        
        viewed as an extension of the conservation of charge
        
        since charge is the product of current and time the current has been flowing
    - Kirchoff's voltage law
      - approximate the circuit with lumped elements
        
        so that time-varying magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible.
        
        maxwell-faraday equation reveals: $\Delta \times E = - \frac{\delta B}{\delta t} = 0$ in the exterior region
  - incoming and outcoming mass and energy flows of a node must balance out
    - $m_b$ denotes the mass flow of a branch connected to the node
    - $m_n$ denotes the mass flow that enters or leaves the system through the node itself
    - $q_n$ is the heat flow entering or leaving a node
    - $N_b$ is the number of branches connected
    - subscripts $in$ and $out$ denote incoming and outgoing flows
  - a perfect mixture is assumed at each node
    - the temperature of all outgoing flows equate to the node temperature $\sum\limits_{b=1}^{N_b} m_b + m_n = \sum\limits_{b=1}^{N_b} v_b A_b \rho_b + m_n = 0$ $\sum\limits{b=1}^{N_b}q_n = \sum\limits_{b=1}^{N_b , in} m_b c_p T_b - \sum\limits_{b=1}^{N_b, out} m_b c_p T_n = 0$
    - the pressure or temperature differences within a mesh must sum up to $0$ state below $\sum\limits_{b=1}^{N_b} \Delta p_b = 0$ $\sum\limits_{b=1}^{N_b} \Delta T_b = 0$
  - to solve for the unknown variables of node pressures and branch velocities
    - physical models are implemented for components in pandapipes
      - components include three types
        
        nodes
        
        branches connecting two nodes
        
        node elements, which are connected to single nodes and change their properties
    - the following component models are implemented
      - junctions
        
        the node representations corresponding to the mass and energy flow balanced as formulated
        
        pressure and temperature ned to be preset as boundary conditions
      - external grids
        
        the reference node representation for the connections to other systems
        
        this node element fixes pressure or temperature for the connected junction to form above boundary conditions
      - sinks and sources
        
        node elements which insert a mass flow entering or leaving the connected junctions
      - pipes
        
        main branch components
        
        they connect two junctions
      - valves
        
        branch components
        
        that also disconnect the two connected junctions from each other
      - pumps
        
        branch components
        
        that introduce a pressure lift
        
        calculated according to a characteristic that sets the operating point in dependency of the calculated flow rate
      - circulation pumps
        
        useful for district heating grid calculations

Pipes

Pipe Attributes

$p$, pressure
$x$, length along a pipe
$f$, wall friction factor
$D$, inner pipe diameter
$\rho$, the fluid density
$v$, the fluid velocity
$Re$, the fluid Reynolds number
$k$, the pipe roughness

$m$, Mass Flow Rate
- BlendPath solves for steady-state pressure profile
  - the hydraulics within the pipeline network, and more specifically the pressures
    - are determined by evaluating the momentum balance for a given pipe: $\frac{dp}{dx}=\frac{f}{2D}\rho v |v| - \rho v \frac{dv}{dx}$
      - where $p$ is the pressure
      - $x$ is the length along a pipe
      - $f$ is the wall friction factor
      - $D$ is the pipe inner diameter
      - $\rho$ is the fluid density
      - $v$ is the fluid velocity
      - BlendPath calculates the friction factor using the explicit Hofer approximation of the Colebrook-White formula: $f=(-2\log_10[\frac{4.518}{Re}\log_10(\frac{Re}{7})+\frac{k}{3.71D}])^{-2}$
        
        where $Re$ is the fluid Reynolds number
        
        $k$ is the pipe roughness
  - so the mass flow rate is constant in any singular pipe
  $m = A (\frac{M_W}{ZRT}\frac{D}{fL})^{0.5} \sqrt{p^2_2 - p^2_1}$
  - $A$ is pipe cross-sectional area
  - $M_W$ is the fluid's molecular weight
  - $R$ is the gas constant
  - $T$ is the temperature
  - $L$ is the pipe length
  - $p_2$ $p_1$ are the absolute pressures at the pipe inlet and outlet
  - $Z$ is the gas compressibility, calculated using the average fluid properties in the pipe
    - $Z$ is calculated using average fluid properties in the pipe as: $Z_{avg} = \frac{P_{avg}M_W}{RT_{avg}\rho_{avg}}$
      - where the average temperature, pressure, and density are calculated as: $p_{avg}=\frac{2}{3}(p_1+p_2 - \frac{p_1p_2}{p_1 + p_2})$ $T_{avg} = \frac{T_1 + T_2} {2}$ $\rho_{avg} = \rho (p_{avg},T_{avg})$
        
        where $T_1$ and $T_2$ are the fluid temperatures at the pipe inlet and outlet
        
        the density calculation depends on which equation of state is applied
        
        Papay EOS or Redlich-Kwong EOS
- GasModels implements a steady-state model of gas flowed based on the Weymouth formulation that uses 1-D hydrodynamic equations for natural gas flow in a pipe
  - Given the assumption of steady flow, the conservation of momentum equation for flow of gas in a pipe: $p\frac{\partial p}{\partial x} = -\frac{\lambda a^2 \phi|\phi}{2D}$
    - $p$ is pressure
    - $\lambda$ is a non-dimensional friction factor
    - $\phi$ is mass flux
    - $D$ is the diameter of the pipe $a^2 = \frac{ZRT}{m}$
      - $Z$ is gas compressibility factor
      - $R$ is the universal gas constant
      - $m$ is the molar mass of the gas
      - $T$ is the gas temperature
    - mass conservation reduces to: $\frac{\partial \phi}{\partial x} = 0$
      - we assume pipe area does not change with $x$
      - we assume gas pressure and gas density satisfy the equation of state, i.e. $p = a^2p$
  - Given that $p\frac{\partial p}{\partial x} = \frac{1}{2} \frac{\partial p^2}{\partial x}$ and $\phi$ is a constant throughout the pipe
    - the conservation of momentum equation is is integrated from the start of the pipe at $x = 0$ to the end of pipe at $x = L$
      - where $L$ is the length of the pipe
    - the equation for flux across the pipe is: $p^2(L) -p^2(0) = \frac{-\lambda L a^2 \phi|\phi|}{D}$
  - We express the mass flux through the pipe in terms of mass flow, $f$
    - where $f=\phi A$
      - $A=\frac{\pi D^2}{4}$ is the cross-sectional area of the pipe
      - thus, the equation for mass flow through a pipe is: $p^2(L) -p^{-2}(0) = \frac{-\lambda L a^2 f|f|}{DA^2}$
  - We non-dimensionalize the units to create a better numerically conditioned problem
    - we use a typical pressure $p_0$ and a typical mass flow $f_0$
      - and normalize the equations $p^{-2}(L)-p^{-2}(0) = -\bar{f}|\bar{f}|(\frac{\lambda L}{D})(\frac{f^2_0 a^2}{A^2p^2_0})$
        
        where $\bar{f}=\frac{f}{f_0}$ and $\bar{p}=\frac{p}{p_0}$ are dimensionless mass flow pressure, respectively
        
        and are both of order one
        
        we define resistance, $w$, as the constant $w = (\frac{\lambda L}{D})(\frac{f^2_0 a^2}{A^2 p^2_0})$
  - Finally, in most data sets, nodal injections and withdrawals are defined in terms of volumetric flow, $q$, at a standard temperature and pressure (STP) conditions
    - Given this data, we non-dimensionalize based on $q$
      - at STP conditions , the mass flow is derived as: $f=\frac{q}{\rho_s}$
        
        where $\rho_s$ is the gas density at STP conditions
- Pandapipes implements a Newton-Raphson solver for steady-state anaylsis of fluid dynamics of all components with their hydraulic quantities are calculated
  - i.e. the pressures for all nodes and the velocities for all branches
  - they also provide calculations for the heat transfer through the network and from the network to external components or the environment
  - the Newton-Raphson solver builds an structure containing a system matrix
    - i.e. a system of equations in order to solve for pressure and velocity based on initial guesses and formulations for the derivatives
    - the structure contains all the information for constructing the Jacobian matrix
      - the term "Jacobian" is used for both the matrix of first partial derivatives of a vector-valued function and its determinant
        
        Given a set $y=f(x)$ of $n$ equations in $n$ variables $x_1,..., x_n$ $y= \left[ \begin{array}{cc} f_1(x) \\ f_2(x) \\ ... \\ f_n(x) \end{array} \right]$ or $\left\{ \begin{array}{rcl} y_1 = f_n(x_1,...,x_n) \\ ... & ... \\ y_n = f_n(x_1,...,x_n) \end{arry}\right$
        
        the Jacobian matrix $\mathbf{J}(x_1,...,x_n) = \left [ \begin{array}{ccc} \frac{\partial y_1}{\partial x_n} & \cdots & \frac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial y_n}{\partial x_i} & \cdots & \frac{\partial y_n}{\partial x_n} \end{array}\right]$
        
        the determinant of $\mathbf{J}$ is the jacobian determinant $J = |\frac{\partial(y_1,...,y_n)}{\partial(x_1,...,x_n)}|$
      - The Pandapipe internal table (structure) contains values like the load vectors and their derivatives
        
        only really active parts of the network are considered
      - The residual and its derivatives with respect to the estimated values $(p,v,T)$ are calculated and stored in branch pandapipes internal table
      - The system matrix is constructed
        
        all the derivatives are written into one large sparse matrix
        
        in which the row indices represent the node indices
        
        followed by the branch indices and the column indices represent the indices of the solution variables
        
        typicall they also belong to nodes and brancehs
        
        it contains all the derivatives
        
        the load (residual) vector is constructed by summarizing all the node-related residuals from the branch pandapipes internal table at all nodes
        
        e.g. incoming and outgoing mass flows
        
        and appending the residuals calculated for the branches to it
      - The linearized system of equations for the hydraulic magnitudes
        
        the Jacobian matrix is multiplied with the vector with estimates' step outputting a load vector
    - Newton-Raphson solver is implemented as followed:
      - initial guesses for $p$ and $v$ are inserted in the node and branch tables for each of the components
        
        initial guesses are derived from where all values are the same or derived from values in the tables
        
        i.e. nominal pressure level of a network
        
        or taken from structure if previous calculations have been stored
      - As long as the stopping criteria…
        
        are not reached
        
        usually maximum iterations as well as residual, pressure and velocity tolerances
        
        each component calculates the residual between given values
        
        such as the mass flow balance for junctions and the pressure difference for pipes
        
        and the calculated values based on the current guesses of $p$ and $v$
        
        also the derivatives of $p$ and $v$ are calculated
      - some boundary conditions have to be given:
        
        the pressure value for at least one network node
        
        which is then the slack node
        
        the mass flow that is extracted at every node by sinks or fed-in by sources
        
        the environmental conditions,
        
        such as the height of each junction, the ambient pressure, the fluid temperature and other fluid properties
        
        similar to the heat transfer calculation
        
        the transferred heat is calculated on the basis of the already calculated mass flow and the magnitudes that are solved for are just temperature values
        
        Here the node temperatures represent the mixed fluid temperatures
        
        And the branch temperatures represent the temperatures at internal nodes at the rear of the branches
        
        the mixed fluid temperature is calculated based on all incoming flows to the node
        
        internal node temperature depends on the previous node's temperature and the heat losses to the environment
    - In pandapipes a procedure is used to find hydraulic unknown variables $p_n$ for nodes and $v_b$ for branches
      - start with initial guess
        
        derivatives of the node and branch equations for the unknown variables at the current guess are determined to find the next Newton step
        
        $\Delta v_b$, $\Delta p_{n_1}$ and $\Delta p_{n_2}$ denote the Newton step to be found
        
        for the branch velocities and respective pressures of the two nodes connected to each branch
        
        $\Delta p_b$ denotes the pressure drop along the branch
        
        $m_b$ denotes the mass flow through the branch
    $\sum\limits_{b=1}^{N_b} \frac{\partial m_b}{\partial v_b} \cdot \Delta v_b = m_n$
$\frac{\partial \Delta p_b}{\partial p_{n_1}} \cdot \Delta p_{n_1} + \frac{\partial \Delta p_b}{\partial p_{n_2}} \cdot \Delta p_{n_2} + \frac{\partial \Delta p_b}{\partial v_b} \cdot \Delta v_b = \Delta p_b$
- Node components implement a mass flow or power balance
  - the mathematics behind the component models in pandapipes how they are setup into a system of nonlinear equations
    - presented equations correspond to the pipe component model but also are applicable to other branch components considering respective characteristics
      - e.g. valves have a length of 0, which means the Darcy friction term is neglected and only a lumped pressure is introduced
      - for an incompressible medium, the pressure variation along a pipe section with length $dl$ can be described as: $dp = -\frac{\rho \cdot v^2}{2}\cdot (\frac{\lambda}{d}\cdot dl + \gamma) + \rho \cdot g \cdot dh$
        
        pressure difference described with the terms on the RHS of the equation
        
        pressure loss due to friction is proportional to the Darcy friction factor $\lambda$
        
        it also depends on the pipe diameter $d$
        
        a lumped pressure loss can be expressed using the pressure loss coefficient $\zeta$
        
        this pressure loss is introduced
        
        e.g. by bends or components which are not represented by a component model
        
        the pressure losses due to both friction and lumped components are proportional to the dynamic pressure $\frac{\rho-v^2}{2}$ where $\rho$ is the fluid density and $v$ is the flow velocity
        
        pressure difference is introduced by a given height difference $\Deltah$ between connected junctions is given by the last term
        
        the difference depends on the fluid density and the gravitational acceleration $g$
      - in the case of a compressible media, the density of the medium is expressed by using the ideal gas law and a reference state $\frac{p}{\rho \cdot T} = \frac{pN}{\rho N \cdot T_N}$ $\rightarrow \rho = \rho_N \cdot \frac{p \cdot T_N}{p_N \cdot T}$
        
        the reference state, noted with index $N$, corresponds to the known gas normal properties
        
        to add real gas behavior, it is modified by a compressibility factor
      - For the reference state, a velocity can be found with the help of relation for the mass flow, $m$, inside the pipe
        
        the system velocity can be formulated as a function of the velocity of the reference state $v_N$ $m = \rho \cdot A \cdot v = \rho_N \cdot A \cdot v_N$ $v = \frac{p_N \cdot T \cdot v_N}{p \cdot T_N}$ $\frac{dp}{dl} = - \lambda\frac{\rho_N \cdot v^2_N}{2 \cdot d} \cdot \frac{p_N}{p} \cdot \frac{T}{T_N}$
        
        capures the continouos velocity and pressure distributions along a pipe

Compressors

In BlendPATH's pipeline network model, compressors are represented as a branch connecting the compressor's inlet and outlet nodes
- modeling compressors to estimate compressor shaft work and to determine shaft compression station capacity
- model inputs include gas mixture's thermodynamic properties at the compressor inlet and outlet and the mass flow rate of the gas being compressed
- the mass-basis enthalpy, $h_1$, and entropy, $s_1$, at the inlet of the compressor are estimated via Cantera, python library, as functions of temperature, pressure, and composition $h_1 = f(T_1, p_1, X_1)$ $s_1 = f(T_1, p_1, X_1)$
  - where $f$ denotes the thermodynamic property evaluation within Cantera
  - and $T_1, p_1$ and $X_1$ are the temperature, pressure, and molar composition at the compression inlet
  - conversely, the enthalpy at the compressor outlet, $h_{2,s}$
    - for an isentropic process is given as: $\Delta h_{isentropic} = h_{2,s} - h_1$
      - the real change in enthalpy, $\Delta h_{real}$, is calculated by dividing the compressors's adiabatic head with an isentropic efficiency:
      $\Delta h_{real} = \frac{\Delta h_{isentropic}}{\eta compressor}$
      - where $\eta_{compressor}$ is the isentropic efficiency for compression
        
        compressor stations assume an isentropic efficiency of 0.78 for gas-driven compressors and 0.88 for eletricity-driven compressors
      - compressor shaft power, $W_{shaft}$ is calculated as: $W_{shaft} = \Delta h_{real} m_{compressor}$
        
        where $m_{compressor}$ is the mass flow rate of fluid exiting the compressor
        
        assuming that any gas fueling the compression station is extracted prior to the gas entering the compressor
      - the driver fuel rate $W_{fuel}$ is given as: $W_{fuel}=\frac{W_{shaft}}{\eta_{driver}}$
        
        where $\eta_{driver}$ is the driver efficiency for a gas-driven compressor
        
        compressor stations assume a driver efficiency of 0.357
        
        for electricity-driven compressor stations, the default driver efficiency is calculated as: $\eta_{driver} = 8E-5x^4 - 0.0015x^3+0.0061x^2+0.0311x+0.7617$
        
        where $x = ln(W_{shaft})$, and $W_{shaft}$ is in units of kilowatts
In GasModels

Pandapipes does not model compressors?

$m$, Mass Flow Rate
$\{\underline{\alpha},\bar{\alpha}\}$ Compression Ratio
- time-dependent compression ratio between intake and outlet pressure
Directionality