Unit 4: Flow in constant area duct

I. Orientation — One-dimensional compressible duct flow

Constant-area duct flow describes how a compressible fluid changes when friction or heat transfer acts without a geometrical area change. Fanno flow isolates wall friction in an adiabatic duct, whereas Rayleigh flow isolates heat transfer in a frictionless duct; both models predict a limiting sonic state.

• Common assumptions:
  • Steady, quasi-one-dimensional flow through a duct of constant area (A).
  • Perfect gas with constant specific heats (c_p), (c_v), and ratio (\gamma=c_p/c_v).
  • Uniform properties across each section; changes occur only along axial coordinate (x).
  • No shaft work and negligible changes in potential energy.
• Common notation: (p,T,\rho,u) are static pressure, temperature, density, and velocity; (p_0,T_0) are stagnation values; (M=u/a) is Mach number; (a=\sqrt{\gamma RT}) is acoustic speed; and (R) is the gas constant.
• Sonic reference state: A superscript (*) denotes the state at (M=1) on the same Fanno or Rayleigh line, not necessarily the local duct condition.
• Choking principle: Friction or heating can drive either a subsonic or supersonic stream toward (M=1). Once sonic conditions occur, prescribed upstream conditions cannot sustain a larger mass flow through the same duct.

II. Fanno Flow — Adiabatic flow with wall friction

A. Fanno flow

Fanno flow is steady, adiabatic flow of a perfect gas through a constant-area duct in which wall friction is the only irreversibility.

• Model conditions: (A) is constant, heat transfer is zero ((\delta q=0)), and wall shear stress (\tau_w) is nonzero.
• Energy principle: With no heat or work, stagnation enthalpy remains constant:
  
  TEXT

  h₀ = h + u²/2 = constant
  T₀ = T[1 + (γ − 1)M²/2] = constant

  
  Here (h=c_pT) and (h_0=c_pT_0).
• Irreversibility: Friction generates entropy, so (ds>0), while stagnation pressure (p_0) decreases downstream.
• Fanno line: On an enthalpy–entropy diagram, all states having the same mass flux and stagnation enthalpy form a Fanno line. Its maximum-entropy point is the sonic state.
• Direction of change:
  1. Subsonic flow: Friction accelerates the stream toward (M=1).
  2. Supersonic flow: Friction decelerates the stream toward (M=1).

B. Fanno flow equations and solutions

The Fanno equations relate any local state to the sonic reference state and determine the permissible duct length before choking.

• Conservation equations:
  
  TEXT

  ρuA = constant
  h + u²/2 = constant
  dp + ρu du + (4τw/D)dx = 0

  
  Here (D) is hydraulic diameter and (dx) is an elemental duct length.
• Dimensionless property relations:
  
  TEXT

  T/T*   = (γ + 1)/[2 + (γ − 1)M²]
  
  p/p*   = (1/M)√{(γ + 1)/[2 + (γ − 1)M²]}
  
  ρ/ρ*   = (1/M)√{[2 + (γ − 1)M²]/(γ + 1)}
  
  u/u*   = M√{(γ + 1)/[2 + (γ − 1)M²]}

• Stagnation-pressure relation:
  
  TEXT

  p₀/p₀* = (1/M)
            {[2 + (γ − 1)M²]/(γ + 1)}^[(γ + 1)/(2(γ − 1))]

  
  Since (T_0) is constant, entropy approaches its maximum according to
  
  TEXT

  s* − s = R ln(p₀/p₀*)

• Length solution: Using the Fanning friction factor (f), the remaining dimensionless length to the sonic state is
  
  TEXT

  4fL*/D = (1 − M²)/(γM²)
         + (γ + 1)/(2γ)
           ln{[(γ + 1)M²]/[2 + (γ − 1)M²]}

  
  Here (L^) is the length required for the local state to reach (M=1). If the Darcy factor (f_D=4f) is used, the left side becomes (f_DL^/D).
• Two-station solution:
  
  TEXT

  4fL/D = F(M₁) − F(M₂)

  
  (F(M)) denotes the right-hand length function, (L) is station spacing, and subscripts 1 and 2 identify inlet and outlet.

C. Variation of flow properties

Wall friction changes static properties differently on the two Mach-number branches, although entropy always rises downstream.

1. Subsonic branch, (M<1):
   • Velocity and Mach number: Both increase toward their sonic values.
   • Static state: (p), (T), and (\rho) decrease because expansion accompanies acceleration.
   • Stagnation state: (T_0) remains constant, but (p_0) decreases.
2. Supersonic branch, (M>1):
   • Velocity and Mach number: Both decrease toward sonic conditions.
   • Static state: (p), (T), and (\rho) increase as the flow decelerates.
   • Stagnation state: (T_0) remains constant and (p_0) again decreases.
   • Entropy limit: Both branches terminate at (M=1), where entropy is maximum on that Fanno line.
   • Mass-flow consequence: Once the exit becomes sonic, increasing duct length or friction cannot preserve the original inlet state and mass flow; upstream conditions must adjust.

D. Variation of Mach number with duct length

Mach number changes monotonically toward unity as the accumulated friction parameter (4fx/D) increases.

• Differential relation:
  
  TEXT

  [(1 − M²)/(γM²{1 + (γ − 1)M²/2})] · d(M²)/M²
  = 4f dx/D

• Sign interpretation:
  • For (M<1), the coefficient is positive, so (d(M^2)>0).
  • For (M>1), the coefficient is negative, so (d(M^2)<0).
• Finite-length calculation: Given (M_1), evaluate (F(M_1)), subtract (4fL/D), and solve (F(M_2)=F(M_1)-4fL/D) on the same subsonic or supersonic branch.
• Maximum length: The greatest permissible length is obtained by setting (M_2=1):
  
  TEXT

  Lmax = [D/(4f)]F(M₁)

  
  A proposed longer duct is incompatible with the specified inlet state under the ideal Fanno model.

E. Tables and charts for Fanno flow

Fanno tables and charts provide numerical values of sonic-reference ratios and the length function for selected (M) and (\gamma).

• Typical columns: (T/T^), (p/p^), (\rho/\rho^), (u/u^), (p_0/p_0^), and (4fL^/D).
• Reading a table: Select the correct (\gamma), locate the known Mach number, and use its ratios to calculate starred quantities or a second station.
• Length use: Subtract tabulated (4fL^*/D) values between stations; do not use the inlet value alone unless the outlet is sonic.
• Chart interpretation: Property ratios meet at unity when (M=1), while the remaining-length parameter falls to zero.
• Branch control: Numerical inversion of the length function can yield subsonic and supersonic candidates; the physical inlet branch determines the valid solution.

III. Rayleigh Flow — Frictionless flow with heat transfer

A. Rayleigh line

A Rayleigh line represents constant-area, frictionless flow in which heat transfer changes stagnation enthalpy while mass and momentum flux remain constant.

• Model conditions: Wall friction is neglected, (A) is constant, and heat addition or rejection is permitted.
• Mass conservation:
  
  TEXT

  G = ρu = constant

  
  Here (G) is mass flux per unit area.
• Momentum equation:
  
  TEXT

  p + ρu² = p + G²v = constant

  
  Here (v=1/\rho) is specific volume.
• Geometric meaning: In the (p)-(v) plane, (p+G^2v=\text{constant}) is a straight Rayleigh line with slope (-G^2).
• Thermodynamic direction: Heat addition generally drives both subsonic and supersonic flows toward (M=1), where entropy and stagnation temperature are maximum on the line.

B. Rayleigh flow equations

Rayleigh equations connect the local state to the sonic reference state through mass, momentum, energy, and the perfect-gas law.

• Energy equation:
  
  TEXT

  q₂ − q₁ = h₀₂ − h₀₁ = cp(T₀₂ − T₀₁)

  
  Here (q_2-q_1) is heat added per unit mass between stations.
• Static-property ratios:
  
  TEXT

  p/p* = (1 + γ)/(1 + γM²)
  
  ρ/ρ* = (1 + γM²)/[(1 + γ)M²]
  
  u/u* = (1 + γ)M²/(1 + γM²)
  
  T/T* = (1 + γ)²M²/(1 + γM²)²

• Stagnation-temperature ratio:
  
  TEXT

  T₀/T₀* =
  (γ + 1)M²[2 + (γ − 1)M²]/(1 + γM²)²

• Stagnation-pressure ratio:
  
  TEXT

  p₀/p₀* = [(γ + 1)/(1 + γM²)]
            {[2 + (γ − 1)M²]/(γ + 1)}^[γ/(γ − 1)]

• Entropy relation:
  
  TEXT

  s − s* = cp ln(T/T*) − R ln(p/p*)

  
  The result is nonpositive because (s^*) is the maximum entropy on a Rayleigh line.

C. Variation of flow properties in Rayleigh flow

Heat transfer controls the movement along a Rayleigh line, but static temperature does not vary monotonically over the entire subsonic branch.

1. Subsonic heat addition:
   • Mach and velocity: (M) and (u) increase toward sonic values.
   • Pressure and density: Both decrease.
   • Temperatures: (T_0) increases to its sonic maximum; static (T) reaches a maximum at (M=1/\sqrt{\gamma}), then decreases slightly before (M=1).
   • Irreversibility: Entropy increases and (p_0) decreases toward (p_0^*).
2. Supersonic heat addition:
   • Mach and velocity: Both decrease toward (M=1).
   • Static properties: (p), (\rho), and (T) increase.
   • Stagnation properties: (T_0) rises while (p_0) falls toward its sonic-reference value.
   • Heat rejection: Cooling reverses these trends, moving subsonic flow to lower (M) and supersonic flow to higher (M).
   • Thermal choking: The maximum permissible heat addition is
     
     TEXT

     qmax = cp(T₀* − T₀)

     
     Additional heating cannot be accommodated without an upstream-state or mass-flow adjustment.

D. Tables and charts for Rayleigh flow

Rayleigh tables and charts avoid repeated solution of the nonlinear sonic-reference equations.

• Typical entries: (p/p^), (T/T^), (\rho/\rho^), (u/u^), (T_0/T_0^), and (p_0/p_0^) for specified (M) and (\gamma).
• Two-station method: Use the known (M1) to determine starred quantities, apply the heat balance to obtain (T{02}), and then find (M2) from (T{02}/T_0^*).
• Heat evaluation:
  
  TEXT

  q₂ − q₁ = cpT₀*[T₀₂/T₀* − T₀₁/T₀*]

• Chart features: All starred ratios equal unity at (M=1); (T/T^) peaks at (M=1/\sqrt{\gamma}), whereas (T_0/T_0^) peaks at (M=1).
• Branch selection: A stagnation-temperature ratio below unity may correspond to both a subsonic and a supersonic Mach number, so the initial flow regime and heating direction must identify the physical solution.