Pressure evolution in an engine intake manifold (thermodynamics)

Gtonio

Hi,

I am using a model that estimates pressure (P) in an intake manifold. I think there is a mistake in its equations but I cannot find it.

To simplify the problem we can make the following assumptions:
- Only air fills the manifold: air comes into the manifold through the throttle (mass flow rate = mf_thr) and is pumped by the engine through the inlet valve (mass flow rate = mf_vlv).
- Heat transfer is neglected: adiabatic manifold.
- Air expansion through throttle is isenthalpic: if upstream throttle air temperature is a constant, then manifold temperature is a constant (T).

Ideal gas law states: P = (R.T/V).m, with m = manifold air mass and V = manifold volume. With above assumptions (T = constant) and mass conservation law:
dP/dt = (R.T/V).dm/dt = (R.T/V).(mf_thr – mf_vlv) {A}

However, model equations are as follow:
- First principle of thermo for open systems (no heat transfer and no work): dU/dt = mf_thr.h_thr – mf_vlv.h_ vlv [0], with h = specific enthalpy, U = internal energy of air in intake manifold
- Internal energy and enthalpy of an ideal gas: U = m.Cv.T, h = Cp.T, with Cp and Cv assumed to be constants on the considered temperature range.
- Derivative of U = m.Cv.T: dU/dt = Cv . (dm/dt . T + dT/dt . m) [1]
- Derivative of ideal gas law P = m.R.T/V: dP/dt = R/V . (dm/dt . T + dT/dt . m) [2]
- [1] and [2] give: dP/dt = R/(Cv.V) . dU/dt [3]
- [3] and [0] give: dP/dt = R/(Cv.V) . (mf_thr.h_thr – mf_vlv.h_ vlv)
=> dP/dt = R/(Cv.V) . (mf_thr.Cp.T_thr – mf_vlv.Cp.T_vlv), with T_thr = temperature at throttle level, T_vlv = temperature at inlet valve level
=> dP/dt = R.Gamma/V . (mf_thr.T_thr – mf_vlv.T_vlv), with Gamma = Cp/Cv

If T_thr = T_vlv = T (manifold temperature), then:
dP/dt = (R.T.Gamma/V).(mf_thr – mf_vlv) {B}

Equation [B] contains Gamma coefficient, which is not present in equation [A]. Where is the mistake?

Thanks in advance for helping me to understand this tricky problem.