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 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.

 

[Chestermiller]

The first analysis assumes that the temperature is constant, which is incorrect because expansion cooling and compression heating of the gas within the manifold. The second analysis does take into account the temperature variation, but the analysis is incorrect for other reasons.