Flame Transfer Function

The effect of unsteady heat release fluctuations on the acoustic pressure are modeled using a flame response model. Simcenter STAR-CCM+ provides the N-Tau Model or the User N-Tau Model. These models use a time delay to account for the finite time between a heat release oscillation and the response of the pressure oscillation.

A common mechanism of thermo-acoustic instability in combustors is as follows. A heat release fluctuation at the flame creates an acoustic pressure wave, which propagates upstream and modulates the fuel flow rate. This causes an equivalence ratio fluctuation, which convects downstream to the flame zone and results in a heat release fluctuation—and this cycle continues. The effect of the unsteady combustion on the acoustic pressure is hence delayed by approximately the convection time from the inlet to the flame. Other mechanisms of instability are similarly characterized by time-delays.

Simcenter STAR-CCM+ models the Flame Transfer Function (that is, the effect of unsteady heat release on the acoustic field) using an n-tau model, where n is a multiplier (called an interaction index), and tau is the specified time delay.

N-Tau Model

The Simplified N-Tau Model calculates the combustion heat release source by:

Figure 1. EQUATION_DISPLAY

\hat{q} (x) = \frac{q_{t o t}}{i ω ρ_{0} (x_{r e f}) U_{b u l k}} n_{u} (x) e^{i ω τ (x)} \nabla \hat{p} (x_{r e f}) \cdot n_{r e f}

(4790)

where

q_{t o t}

is the total heat release,

i

is the imaginary number

\sqrt{- 1}

ω

is the acoustic frequency (eigenvalue),

ρ_{0}

is the mixture density,

x_{r e f}

is a reference location typically at the burner inlet,

U_{b u l k}

is the bulk velocity,

n_{u} (x)

is the interaction index,

τ (x)

is the time lag,

\hat{p}

is the mean pressure, and

n_{r e f}

is the reference unit vector which indicates the direction from the reference location to the flame (as shown in the diagram above).

The interaction index $n_{u} (x)$ corresponds to the amplitude of the flame response $n$ at location $x$ .

Based on findings from Nicoud F. et al [742], assuming that

n_{u} (x)

is constant within the flame region—where the heat-release rate is non-zero:

Figure 2. EQUATION_DISPLAY

n_{u} (x) = η = \frac{n}{δ_{f}} \frac{U_{b u l k}}{q_{t o t}} \frac{γ p_{0}}{γ - 1}

(4791)

elsewhere, where the heat-release rate is zero:

Figure 3. EQUATION_DISPLAY

n_{u} (x) = 0

(4792)

where

η

is the constant value of the thermal efficiency within the flame,

δ_{f}

is the flame thickness,

γ

is the adiabatic heat capacity ratio coefficient (

\frac{C_{p}}{C_{v}}

p_{0}

is the reference pressure, and

n

is a user-defined scalar constant.

In Simcenter STAR-CCM+, Eqn. (4791) is multiplied by a scaled heat release rate:

Figure 4. EQUATION_DISPLAY

n_{u} (x) = η = \frac{n}{δ_{f}} \frac{U_{b u l k}}{q_{t o t}} \frac{γ p_{0}}{γ - 1} {\dot{q}}_{s c a l e d} (x)

(4793)

where:

Figure 5. EQUATION_DISPLAY

{\dot{q}}_{s c a l e d} (x) = (\frac{\dot{q} (x) - {\dot{q}}_{\min}}{{\dot{q}}_{\max} - {\dot{q}}_{\min}})

(4794)

Simcenter STAR-CCM+ calculates the combustion heat release source for the simplified N-Tau model assuming a specified constant time-delay

τ (x) = τ

Figure 6. EQUATION_DISPLAY

\hat{q} (x) = \frac{n}{δ_{f}} {\dot{q}}_{s c a l e d} (x) \frac{γ p_{0}}{i ω ρ_{0} (x_{r e f}) (γ - 1)} e^{i ω τ} \nabla \hat{p} (x_{r e f}) \cdot n_{r e f}

(4795)

The flame thickness

δ_{f}

is calculated by:

Figure 7. EQUATION_DISPLAY

δ_{f} = {[\int {\dot{q}}_{s c a l e d} d V]}^{1 / 3}

(4796)

Simcenter STAR-CCM+ obtains the flame response matrix from the above expressions as follows:

Figure 8. EQUATION_DISPLAY

D (ω) \hat{P} = \int i ω (γ - 1) \hat{q} d V

(4797)

User N-Tau Model

The (generalized) User N-Tau model calculates the combustion heat release source as follows:

Figure 12. EQUATION_DISPLAY

\hat{q} (x) = \frac{q_{t o t}}{i ω ρ_{0} (x_{r e f}) U_{b u l k}} n_{u} (x) e^{i ω τ (x)} \nabla \hat{p} (x_{r e f}) \cdot n_{r e f}

(4801)

in which

n_{u} (x)

is the user-specified full interaction index field and

τ (x)

is the user-specified entire time delay field.

$x_{r e f}$ is the reference location, $n_{r e f}$ is the reference direction, and $U_{b u l k}$ is the bulk velocity.