3.0 Introduction
The main objective of an engineering analysis is to have the prediction results as close to the physical reality as possible. These results are said to be numerically accurate if the modeling equations are solved correctly. They are considered physically accurate if they also correctly predict the physical reality, which is independent of the method of analysis. In general, for prediction results to be physically accurate, they ought to be numerically accurate. In the context of computer code (design tool) development, the numerical accuracy of the computed results is ensured through code verification, and their physical accuracy is determined by code validation. In our discussion, we will tacitly assume that the computational method used for solving a flow network ensures numerical accuracy. The accuracy in the present context essentially means physical accuracy.
The physics-based one-dimensional (1-D) thermofluids modeling of various components of internal flow systems of a gas turbine is a design necessity, offering the best compromise of prediction accuracy, speed, and cost. Predictions are generally made using complex flow networks of these components in all three design phases: conceptual, preliminary, and detailed. The reliability of design predictions depends in large part on the company-proprietary empirical correlations. Each robust design is performed with compressible flow networks in a short available design cycle time. The CFD technology is leveraged in two ways; first, to develop a better understanding of the component flow physics, reinforcing its 1-D modeling, and second, to delineate areas of design improvement in the detailed design phase using an entropy map. In this chapter, we limit our discussion to steady compressible flow networks with the possibility of internal choking and normal shocks.
The accuracy of results from a flow network primarily depends upon two factors: (1) the core formulation that ensures that the conservation laws of mass, momentum, energy, and entropy are duly satisfied for each flow element and junction in the network and (2) the empirical correlations, which are based on the benchmark quality data, are representative of the physical reality. While there is always a need for improved empirical correlation for an existing or newly designed element, the core mathematical formulation based on the established laws of flow and heat transfer physics ought to remain invariant. Accordingly, the physics-based modeling in the context of our discussion in this chapter will imply that all the conservation laws are fully satisfied in the flow network. Thus, any lack of accuracy of a network solution can be entirely attributed to the deficiencies in one or more empirical correlations used in the network.
3.1 1-D Flow Modeling of Components
The physics-based 1-D flow modeling of each component of gas turbine internal flow systems is achieved through large control volume analysis of the conservation equations of mass, momentum, energy, and entropy. For a compressible flow, the momentum equation remains coupled with the energy equation through density, which is computed using the equation of state of a perfect gas. In this chapter, we present the modeling of a duct and an orifice, which are two basic components of an internal flow system. Modeling of other special components, such as vortex, rotor-rotor and rotor-stator cavities, and seals are presented in the following chapters.
3.1.1 Duct with Area Change, Friction, Heat Transfer, and Rotation
In this section, we present the most general modeling of one-dimensional compressible flow in a variable-area duct with wall friction, heat transfer, and rotation (constant angular velocity) about an axis different from the flow axis. The modeling methodology also allows for the presence of internal choking (M = 1M=1) and normal shock, which features abrupt changes in flow properties. In the present approach, the long duct is divided into multiple control volumes, which are serially coupled such that the outlet flow properties of one control volume become the inlet flow properties for the downstream one. In each control volume, wall boundary conditions are assumed uniform, although they may vary over different control volumes. Thus, in order to model the entire duct, we need to develop modeling equations only for a representative control volume, as has been done in the following sections.
3.1.1.1 Mass Conservation (Continuity Equation)
For the duct control volume shown in Figure 3.1, the velocity, density, and area for the x-direction flow are assumed to vary linearly from inlet (section 1) to outlet (section 2). For a steady flow through the control volume, the mass conservation yields
(3.1)where
In terms of the total pressure, total temperature, Mach number, and total-pressure mass flow function (presented in Chapter 2), we can compute mass flow rate at the duct control volume inlet and outlet as
3.1.1.2 Linear Momentum Equation
For the duct control volume shown in Figure 3.1a, we can write the steady flow momentum equation as
(3.2)where
FP≡FP≡ Pressure force acting on the control volume in the momentum direction
Fsh≡Fsh≡ Shear force acting on the control volume opposite to the momentum direction
Frot≡Frot≡ Rotational body force acting on the control volume in the momentum direction
Let us now evaluate the surface forces due to static pressure and wall shear and the body force due to rotation.
Pressure force (FPFP). Figure 3.1b shows the static pressure distribution on the duct control volume, where we have assumed that the average pressure on the lateral surfaces is the average of inlet and outlet pressures. The total pressure force in the x direction, resulting from the surface pressure distribution, can be expressed as
(3.3)Equation 3.3 shows that the net force resulting from a pressure distribution on the duct control volume with area change in the flow direction is the product of the difference in the inlet and outlet pressures and the mean flow area.
Shear force (FshFsh). The net shear force acting on the control volume in the momentum direction can be expressed as
(3.4)where f¯
is the average value of the Darcy friction factor over the lateral control volume surface.
Rotational body force (FrotFrot). The component of the rotational body force acting on the control volume in the momentum direction can be expressed as
(3.5)Substituting the foregoing expressions for the pressure force, shear force, and rotational body force in Equation 3.2, we obtain
(3.6)Thus, knowing Ps1Ps1 at the inlet, we can use Equation 3.6 to compute the static pressure Ps2Ps2 at the outlet as
In Equation 3.7, the terms representing changes in static pressure due to friction, rotation, and momentum-change are given as follows:
(3.8)
(3.9)
(3.10)3.1.1.3 Energy Equation
The steady flow energy equation for the duct control volume involves heat transfer at the wall and work transfer due to rotation. Both of these energy exchanges result in change in gas total temperature from inlet to outlet. It is important to note that, while the rotational work transfer does not depend upon the simultaneous heat transfer, the heat transfer does depend upon the simultaneous work transfer. This work transfer changes the gas total temperature and hence the difference between the wall temperature and the adiabatic wall temperature, affecting heat transfer. Sultanian (2015) provides a closed-form analytical solution for the coupled heat transfer and rotational work transfer in a compressible duct flow, including a correction term if one decides to simply add the separately-computed changes in gas total temperature due to heat transfer and rotation. Thus, the gas total temperature at the exit of the flow element with rotation can be computed as
where
(ΔTtR)HT≡ΔTtRHT≡ Change in gas total temperature due to heat transfer
(ΔTtR)rot≡ΔTtRrot≡ Change in gas total temperature due to rotation
(ΔTtR)CCT≡ΔTtRCCT≡ Heat transfer and rotational work transfer coupling correction term
Heat transfer: (ΔTtR)HTΔTtRHT. For the convective heat transfer, we assume that the duct control volume walls are isothermal (constant wall temperature TwTw) and the heat transfer coefficient h, which is either specified or computed from an empirical correlation, remains constant from inlet to outlet. We further assume that the adiabatic wall temperature at each section of the control volume equals the gas total temperature, implying a recovery factor of 1.0.
Based on the analysis presented in Chapter 2, the change in gas total temperature due to heat transfer from inlet to outlet can be expressed as
where
Rotational work transfer: (ΔTtR)rotΔTtRrot. When the gas enters a rotating duct, it assumes the state of solid body rotation, that is, the gas starts rotating at the constant angular velocity of the duct. Under adiabatic conditions, the change in gas total temperature (relative to the control volume rotating at constant speed) between two radial locations can be obtained by equating gas rothalpy (see Chapter 2) at these locations. Thus, we can write
(3.13)Coupling Correction Term: (ΔTtR)CCTΔTtRCCT. Sultanian (2015) derived the heat transfer and rotational work transfer coupling correction term as
(3.14)3.1.1.4 Internal Choking and Normal Shock Formation
Compressible flow in a variable-area duct, for example in a convergent-divergent nozzle, can feature internal choking at a section where the flow velocity equals the local speed of sound (M = 1M=1). If the flow area increases beyond this section, the flow becomes supersonic with the possibility of a normal shock if the duct exit conditions are subsonic. The flow properties vary continuously across a section where the flow is choked; however, they vary abruptly across a normal shock. In the modeling of a long variable-area duct, a good way to simulate the choked-flow section is to make it coincide with an interface between adjacent control volumes. For simulating a normal shock, however, it is better to imbed a negligibly thin control volume within which the normal shock occurs. Using the normal shock equations presented in Chapter 2, we can then compute the properties at the outlet of this imbedded control volume.
3.1.1.5 Flexibility of Duct Flow Modeling
The comprehensive 1-D modeling of a duct flow presented in the foregoing is much more versatile that it first appears. An orifice can also be simulated using a short duct with no area change and with specified discharge coefficient CdCd such that the mass flow rate through the orifice (duct) can be computed using the equation
(3.14)In Equation 3.14, the flow properties correspond to the mechanical area A that yields the effective area Aeff = CdAAeff=CdA. The total-pressure mass flow function F̂ft
is a function of Mach number, which is uniquely computed from the isentropic pressure ratio at A. In terms of the static-to-total pressure ratio Ps/PtPs/Pt, we can write Equation 3.14 as
(3.15)which holds good for 0.5283 ≤ Ps/Pt < 10.5283≤Ps/Pt<1 where Ps/Pt = 0.5283Ps/Pt=0.5283 corresponds to the choked flow condition (M = 1M=1) for air with κ = 1.4κ=1.4.
The discrete duct flow modeling presented in this section can also be used to simulate and be validated against the duct flow with each separate effect presented in Chapter 2, namely, isentropic flow with area change (without the friction, heat transfer, and rotation), Fanno flow (without the area change, heat transfer, and rotation), and Rayleigh flow (without the area change, friction, and rotation). We can certainly simulate any combination of various effects on the duct flow. It is interesting to note that a rotating duct flow can also be used to simulate a forced vortex.
3.1.2 Orifice
Orifices are the most ubiquitous element of a gas turbine internal flow system. They are used in both stationary and rotating components either to restrict the flow or to meter it. A choked-flow (M = 1M=1) orifice designed with negligible vena contra can be used as a device with constant mass flow rate, not affected by downstream flow conditions. For constant source and discharge pressures in a passive bleed flow line, orifices in the form of short nozzles are generally used to obtain the desired flow rate. Valves may be modeled as a variable-area orifice.
3.1.2.1 Sharp-Edged Orifice
Figure 3.2a shows a sharp-edged orifice in which the flow through the orifice area at section 2 is driven by the upstream total pressure at section 1 and the downstream static pressure at section 3. Due to flow contraction at the orifice, the flow initially converges to a smaller area, called vena contracta, before expanding to the larger downstream area. The area (AvcAvc) of the vena contracta is a strong function of A3/A2A3/A2, but it does not depend on the pressure ratio across it for an incompressible flow. The overall loss in the total pressure between sections 1 and 3 mainly results from the sudden-expansion loss downstream of the vena contracta. Using the control volume analysis of an incompressible flow, this loss is calculated to be the dynamic pressure of the difference in velocities at the vena contracta and section 3.
Figure 3.2 (a) Sharp-edged orifice and (b) parameters influencing orifice discharge coefficient.
For a compressible flow, there are essentially two approaches to compute mass flow rate through a sharp-edged orifice. The first approach (a classical one) is based on an extension of the incompressible flow method and is given by (see Benedict (1980))
(3.16)where Y is called the adiabatic expansion factor to account for the decrease in density as the flow expands (static pressure decreases) in a compressible flow. Buckingham (1932) proposed the following empirical relation for computing Y:
(3.17)where β is the ratio of orifice diameter to housing pipe diameter. Parker and Kercher (1991) recommended the use of Equation 3.17 for 0.63 ≤ (Ps/Pt)inc ≤ 10.63≤Ps/Ptinc≤1 and 0 ≤ β ≤ 0.50≤β≤0.5, and for 0 ≤ (Ps/Pt)inc ≤ 0.630≤Ps/Ptinc≤0.63, they proposed the following semi-empirical equation:
(3.18)where Y0.63Y0.63 is computed from Equation 3.17 with (Ps/Pt)inc = 0.63Ps/Ptinc=0.63.
To compute the incompressible flow discharge coefficient CdincCdinc used in Equation 3.15, Stolz (1975) proposed the following “universal” equation:
(3.19)where
L1≡L1≡ Dimensionless upstream pressure-tap location with respect to the orifice upstream face
L2≡L2≡ Dimensionless downstream pressure-tap location with respect to the orifice downstream face
Red≡Red≡ Reynolds number based on the orifice diameter
The main motivation behind using the foregoing approach in calculating the mass flow rate of a compressible flow through a sharp-edged orifice is to leverage the vast amount of incompressible flow data available for CdincCdinc. In addition, as discussed next, the method also captures the compressibility effect as the downstream static pressure decreases beyond the nominal choking at the vena contracta, increasing its area with pressure ratio. This results in a higher mass flow rate beyond the critical pressure ratio under the same upstream stagnation conditions.
The second approach, which will be used in this book, to calculating compressible mass flow rate through a sharp-edged orifice is to use Equation 3.15. From Equations 3.15 and 3.16, we can compute the discharge coefficient CdCd as
(3.20)Note that in the numerator of Equation 3.20, 0 ≤ (Ps/Pt)inc ≤ 10≤Ps/Ptinc≤1 while in the denominator we have 0.5283 ≤ Ps/Pt < 10.5283≤Ps/Pt<1, which means the denominator of this equation has a maximum value of 0.6847 that corresponds Ps/Pt = 0.5283Ps/Pt=0.5283.
Figure 3.3a compares the compressible flow CdCd predictions by Equation 3.20 with the measurements of Perry (1949) for β = 0.1β=0.1 over the entire range of the static-to-total pressure ratio Ps/PtPs/Pt. The corresponding linear variation of the adiabatic expansion factor Y is shown in Figure 3.3b, which also shows how Y varies with Ps/PtPs/Pt under isentropic conditions when Cd = Cdinc = 1.0Cd=Cdinc=1.0. Under these conditions, Equation 3.20 yields
(3.21)
Figure 3.3 (a) Comparison of CdCd prediction by Equation 3.20 with the measurements of Perry (1949) for β = 0.1β=0.1 and (b) variation of adiabatic expansion factor Y with static-to-total pressure ratio.
A serious drawback of the method of computing CdCd using Equation 3.20 is that for β>0.5β>0.5 and low pressure ratios it yields Cd>1.0Cd>1.0, which is unacceptable. Parker and Kercher (1991) present a noniterative semi-empirical prediction method to generate intermediate values of CdCd between 0.5959 and 1.0 for 0.5959 ≤ Cdinc ≤ 1.00.5959≤Cdinc≤1.0 over the entire range of Ps/PtPs/Pt. The method, however, has limited experimental validation. High-fidelity CFD, including LES and DNS, may be used to generate CdCd for the entire range of orifice geometry and pressure ratio variations while validating these results against limited available experimental data.
3.1.2.2 Generalized Orifice
Figure 3.2b shows a generalized orifice featuring two geometric parameters, namely, the radius of curvature r/dr/d at the inlet and the length-to-diameter ratio L/dL/d and one inlet flow parameter representing the ratio of the tangential velocity to axial velocity Vθ/VxVθ/Vx. All three parameters, either individually or in combination, significantly affect the orifice discharge coefficient. The maximum benefit of the radius of curvature to minimize flow separation at the orifice inlet and the adverse effect of the vena contracta on discharge coefficient occurs at r/d≈1.0r/d≈1.0. For a thick orifice with L/d≈2L/d≈2, the sudden-expansion loss at the vena contracta is reduced, as a result, the discharge coefficient increases. For L/d>2L/d>2, the adverse affect of the orifice wall friction tends to decrease CdCd. The effect of Vθ/VxVθ/Vx is always to increase flow separation at the inlet causing deterioration in the orifice discharge coefficient. This can occur even with corner radiusing at the orifice inlet.
McGreehan and Schotsch (1988) present a design-friendly method to predict the effects of r/dr/d, L/dL/d and Vθ/VxVθ/Vx on discharge coefficient for a generalized orifice shown in Figure 3.2b. Their approach uses the classical adiabatic expansion factor to generate compressible flow discharge coefficient for a sharp-edged orifice. While the method shows good validation with the available test data for each individual parameter, its validation for the case featuring interaction effects of parameters is rather limited, and the method may at times predict Cd>1Cd>1. Parker and Kercher (1991) further improved the method of McGreehan and Schotsch (1988) by using the total-pressure mass flow function and ensuring that Cd ≤ 1Cd≤1 with a semi-empirical technique.
Existing methods of predicting orifice discharge coefficient for a rotating orifice are generally not reliable due a highly three-dimensional flow field caused by rotation within the orifice. These methods, therefore, tend to overpredict discharge coefficients for a rotating orifice. An obvious effect of rotation is to change Vθ/VxVθ/Vx at the orifice inlet. In addition, if the orifice inlet and outlet radii are appreciably different, the fluid total temperature relative to the orifice will change such that the rothalpy remains constant under adiabatic conditions. According to the isentropic relation, this change in fluid total temperature will results in a change in total pressure. Both these changes will affect the isentropic flow rate through the rotating orifice. Idris and Pullen (2005) present correlations to compute discharge coefficients for rotating orifices. A rotating long orifice with heat transfer and wall friction may alternatively be modeled as a duct, presented in Section 3.1.
From the foregoing discussion it is clear that the need for accurate and dependable 1-D modeling and prediction method for a general compressible flow orifice continues to exist. The lack of benchmark quality measurements in this area remains a serious impediment to further improvement – a situation that is not likely to change in the foreseeable future. Most of the available prediction methods are semi-empirical in nature with limited experimental validation. Another approach to mitigate the current situation is to leverage a CFD-based DOE using all key parameters that influence the discharge coefficient of a generalized orifice and correlating the results in the form a response surface. The response surface correlation can be easily implemented in a flow network code, which is used for modeling internal flow systems of modern gas turbines. The orifice response surface will predict both the direct and interaction effects of various parameters that influence the orifice discharge coefficient.
3.1.3 Vortex
The vortex is an important feature of a gas turbine internal flow system with rotor surfaces. In Chapter 2, we discussed isentropic free vortex, forced vortex, and nonisentropic generalized vortex and their modeling equations to compute pressure and temperature changes across them. The flow in a rotating duct creates a forced vortex that is coupled with other effects of friction and heat transfer. When an internal flow passes over rotor surfaces, like in rotor disk cavities, which we discuss in Chapter 4, it becomes a vortex with concurrent generation of windage.
It is important to note that, unlike a duct or an orifice, the vortex is not a physical component. It is simply a flow feature associated with its bulk circular motion (swirl) about the machine axis of rotation. Accordingly, unlike in a component, knowing the end conditions of a vortex, one may not be able to compute a unique mass flow rate associated with it. Therefore, a vortex will not qualify to be an element (discussed in the next section), which is directly connected to junctions in a flow network. We will henceforth call vortex a pseudo element. For its inclusion in a flow network, it must be sandwiched between two elements, either of which could be a duct or an orifice. The resulting super element, which becomes a mass flow metering component, can be connected to junctions in a flow network.
3.2 Description of a Flow Network: Elements and Junctions
Gas turbine internal flow systems such as blade cooling system, rotor-stator or rotor-rotor cooling system, rotor-stator sealing system, discourager or rim seal system to minimize or prevent hot gas ingestion, inlet bleed heat system, and others are essentially handled using a flow network in which various components (ducts, orifices, seals, vortices, etc.) are modeled as 1-D flow element. Two basic entities of a flow network, shown in Figure 3.4, are element (also called link or branch) and junction (also called chamber or node). An element in this flow network is depicted by a solid line along with an arrow to represent the positive flow direction. A flow network has two types of junctions: internal junctions, depicted by an open circle, and the boundary junctions, depicted by an open square.
Figure 3.4 A flow network.
3.2.1 Element
An element of a flow network typically represents a component of the gas turbine internal flow system. A general flow network may include a variety of elements: duct, orifice, seal, vortex (pseudo component), heat exchanger (super element), and others. As shown in Figure 3.4, each element in the flow network connects two junctions. It is characterized by a mass flow rate, which in steady state remains constant from inlet to outlet. By its very function, an element in a network represents a flow metering device. The element connecting junctions ii and jj is represented by eijeij, which is the same as ejieji. Depending on the flow direction in the element, either of the junctions could be an inlet or an outlet. Note that the state variables at both junctions of an element uniquely determine the actual flow direction in the element. Later in this chapter, we will discuss a physics-based criterion to determine this flow direction. Even if one element in the network is assigned a wrong flow direction, the entire network solution is corrupted, which demonstrates the elliptic nature of the flow field modeled by a flow network, although a 1-D flow is assumed in each element.
Each element in a flow network is analogous to a thermodynamic path connecting two states of a system. All the path variables, evaluated in terms of the amount of work transfer and heat transfer, are associated with the flow through the element. In addition to the geometric parameters, empirical and semi-empirical correlations are specified for each element in the network to determine the friction factor for major loss, discharge coefficient for minor loss, and heat transfer coefficient to compute heat transfer in or out of the flow, etc. Various thermal and hydrodynamic boundary conditions applied to an element are assumed uniform; two- or three-dimensional variations of flow properties within the element are not discernible. These variations, if needed to develop a better understanding of the component, may be obtained through a CFD analysis, which may also be used readily to generate needed data to facilitate 1-D flow modeling of the component in the network to carry out a robust design.
At times, it becomes import to capture variations in geometry and boundary conditions over a flow element connecting two adjacent junctions. Examples include a long pipe line for bleed and coolant supply and the serpentine passage used for internal cooling of turbine airfoils. In such situations, the flow element may be modeled using serially-connected small control volumes without creating additional junctions in the overall network. This modeling practice offers significant economy in the network solution and preserves the dynamic pressure between adjacent control volumes in the flow direction for higher prediction accuracy.
There is often a debate among gas turbine engineers about which pressure, static or total, should be used at the junctions in a flow network to compute mass flow rate through the connecting elements. In steady state, the mass flow rate at any section in an internal flow requires that Pt>PsPt>Ps or Pt/Ps>1Pt/Ps>1, and this mass flow rate remains constant through the element or the gas turbine component that it represents, regardless of how fluid properties change from section to section. As discussed in Chapter 2, we can compute the element mass flow rate at any section using either the total-pressure mass flow function or the static-pressure mass flow function, both of which are functions of the section Mach number and the ratio of gas specific heats. This mass flow rate should be computed at the section, typically outlet, inlet, or where the flow is choked with M = 1M=1. This approach, however, doesn’t determine the flow direction across the section, which must be determined from the entropy change over the element.
It is important to note that the junction pressure at the inlet must be interpreted as the total pressure and that at the outlet as the static pressure. A junction with no associated dynamic pressure behaves like a plenum in which Pt = PsPt=Ps such that, for all flows leaving the junction, the plenum pressure becomes the total pressure, and for all flows entering the junction, the plenum pressure becomes the back pressure. For a subsonic flow through the element, the static pressure at the element exit section must equal the plenum back pressure (static). If, however, the flow is choked at the element exit, its static pressure becomes decoupled from the connected downstream junction, and the element mass flow rate is entirely determine by the upstream conditions. As an example, for the flow network shown in Figure 3.4, consider element e25e25 with the mass flow rate ṁ25
. Even in the presence of dynamic pressure at junction 2 due to the flow along e12e12 and e23e23, the total pressure at the inlet of element e25e25 equals the static pressure at junction 2. Similarly, for a subsonic flow through the element e25e25, its exit static pressure must be equal to the static pressure at junction 5.
3.2.2 Internal Junction
An internal junction connects two or more elements in a network. The flow network shown in Figure 3.4 has two internal junctions, namely, 2 and 5. The internal junction 2, for example, connects elements e12e12, e23e23, and e25e25. In steady state, the continuity equation for this junction yields
(3.22)which, with ṁ21=−ṁ12
, can be written as
(3.23)Thus, at an arbitrary internal junction ii connected through elements to multiple junctions j = 1j=1 to j = kij=ki, as shown in Figure 3.5, we can write the steady continuity equation as
(3.24)where j≠ij≠i. In this equation, we have adopted the convention that ṁij
is nominally positive if the flow direction in the element eijeij is from junction ii to junction jj. According to this convention, we obtain ṁij=−ṁji
.
Figure 3.5 Mass conservation at an internal junction.
To ensure energy conservation at each internal junction, we can compute the mixed mean total temperature of all incoming flows. All outflows take place at this mixed mean total temperature. For example, we can write at junction ii
(3.25)where we have δij = 1δij=1 for ṁij<0
(inflows) and δij = 0δij=0 for ṁij>0
(outflows).
The foregoing discussion makes it clear that the net mass flow rate associated with an internal junction must be zero. All the state variables like pressure and temperature are associated with a junction. While the static pressure is uniform within a junction, the assumption of a uniform total pressure is a matter of modeling assumption depending upon the accuracy and rigor one needs in a design application. In this book, we limit our discussion to dump-type or plenum junctions, where we neglect the dynamic pressure associated with each incoming flow. This is akin to the “tank-and-tube” approach used in modeling a flow network. Sultanian (2015) presents more accurate and detailed modeling of junctions in a compressible flow network.
Thus, we model an internal junction with zero dynamic pressure in it, making the static and total pressures equal. In this case, the junction static temperature also equals the total temperature, which is determined by Equation 3.25 as the mixed mean total temperature of all flows entering the junction. All flows leaving the junction are assumed to be at this total temperature. By contrast, the static pressure of all subsonic flows entering the plenum junction equals the junction pressure, which becomes the total pressure for all flows leaving the junction.
3.2.3 Boundary Junction
A boundary junction in a flow network is either a source or a sink where the boundary conditions are specified. The network solution corresponds to these boundary conditions. Based on the flow directions shown in Figure 3.4, the boundary junctions 1 and 4 are the sources and the boundary junctions 3 and 6 are the sinks. Being sources and sinks, boundary junctions are allowed to violate the steady continuity equation. At a boundary junction, all state properties are generally assumed uniform.
3.3 Compressible Flow Network Solution
A flow network solution yields the mass flow rate through each element and the state properties, such as the total temperature and pressure, at each internal junction. Except for a very simple flow network, hand calculations to obtain a flow network solution could be very tedious and time-consuming. Therefore, the flow networks of gas turbine internal flow systems are typically solved using a computer with the help of a computer code based on a solution method similar to the one presented in this section. Because of nonlinear dependence of an element mass flow rate through on the difference in pressures at the connected junctions, a commonly used solution strategy is to iteratively perform the tasks of element solutions and junction solutions until the continuity equation at each internal junction is satisfied within an acceptable error.
In a flow network, for an assumed initial solution, all the mass flow rates generated in the connected elements will most likely not satisfy the continuity equation at each junction. The continuity Equation 3.24 becomes
(3.26)where Δṁi
is the residual error in the continuity equation at junction ii. For the junction solution, let us assume that the mass flow rate ṁij
through each element eijeij depends only on PsiPsi and PsjPsj. At each iteration, our goal is to change the junction static pressures so as to annihilate Δṁi
in Equation 3.26. Accordingly, we can write
(3.27)Writing Equation 3.27 at each internal junction will result in a system of nonlinear algebraic equations for which no known direct solution method exists. One must therefore solve such a system using an iterative numerical method. At the current iteration, the system is treated as a linear system for which there are many numerical solution methods, for example, the direct method presented by Sultanian (1980); see also Appendix F.
As shown in the foregoing discussion, we can express each element solution in the form ṁij=aijPsi−Psj
, and assuming aijaij as constant at each iteration, we obtain ∂ṁij/∂Psi=aij
and ∂ṁij/∂Psj=−aij
. Substituting them in Equation 3.27 yields
(3.28)