In the model for a rigid tank with an adiabatic isentropic nozzle, how does the function f[M] simplify when the flow is choked (M=1)?
Explanation
The function f[M] is a convenient notation for the combination of terms relating stagnation and static properties. When the flow is choked, the Mach number at the throat is 1, which allows this function to be simplified into a constant determined solely by the specific heat ratio, k. This greatly simplifies the integration of the governing differential equation.
Other questions
In the context of evacuating or filling a semi-rigid chamber, which flow model is considered most appropriate for a relatively fast process in the tube?
What is the physical meaning of the characteristic time, tc, as defined in the general model for evacuating a chamber?
What is the relationship between the reduced temperature and reduced pressure (T_bar and P_bar) in a chamber that undergoes an isentropic process?
When modeling the evacuating of a rigid tank with Fanno flow in the tube, what happens to the entrance Mach number to the tube after the flow becomes chokeless?
For a simple semi-rigid chamber where the volume is linearly related to the pressure (V(t) = aP(t)), what is the physical meaning of the constant 'a'?
In the general model for a semi-rigid chamber where volume is related to pressure by V(t) = aP^n, what physical situation corresponds to n = 0?
When is the Isothermal flow model considered more appropriate than the Fanno flow model for the tube connecting to a chamber?
What is the choking condition (the Mach number where choking occurs) for Isothermal flow in a tube connected to a chamber?
According to the advanced topics section, for what range of entrance Mach numbers (M_in) should the approximate solution Min = sqrt((1 - (P_exit/P0(t))^2) / (k * 4fL/D)) be used?
What is the integrated solution for the reduced pressure P_bar as a function of reduced time t_bar for the filling of a rigid tank from a choked adiabatic isentropic nozzle, assuming k is the specific heat ratio?
When modeling the evacuation of a rigid tank with a choked Fanno flow tube, the solution for the reduced time t_bar is given as t_bar = (2/((k-1)*f[M])) * [P_bar^((1-k)/(2k)) - 1]. What does this equation imply about the relationship between pressure and time?
What is the primary assumption made about the gas process inside the chamber itself in most models presented in this chapter?
How is the dimensionless pressure, P_bar, defined in the general model?
What is the governing equation for the rapid evacuation of a rigid tank with Fanno flow in the tube?
In the general case of a semi-rigid chamber described by V(t) = aP^n, what condition on 'n' is required to obtain a linear relationship between reduced time and reduced pressure during evacuation?
What is the primary reason that Rayleigh flow is not considered applicable for the models in this chapter?
What simplification is made regarding the pressure inside the chamber and the stagnation pressure at the tube's entrance?
In the general differential equation d/dt_bar(P_bar*V_bar/T_bar) +/- P_bar*M_bar*f[M]/sqrt(T_bar) = 0, what does the positive sign signify?
For an isothermal nozzle attached to an isentropic chamber, what is the integrated solution for the reduced pressure P_bar?
A chamber with an initial volume of 1.0 m^3 is connected to a line via a nozzle with a throat area of 0.1 m^2. The gas is at 300K with k=1.4 and R=287 J/kgK. What is the characteristic time, tc, for this system?
What are the two limiting cases used as bounds for the actual process of filling or evacuating a chamber?
The method outlined in Chapters 8 and 9 for solving for the entrance Mach number (M1) is stated to be appropriate for what flow models?
For a completely rigid tank, the relative volume V_bar(t) is equal to what value?
In the context of the 'Simple' General Case (V(t) = aP^n), the author states that in reality the value of n typically lies between what two values?
A chamber is being evacuated through a rubber tube where f=0.025, d=0.01m, and L=0.5m. What is the dimensionless resistance (4fL/D) of the tube?
What is the characteristic time `tc` for an adiabatic isentropic nozzle, as simplified in Equation (11.21)?
When the flow is in a chokeless condition, how many equations must be solved to find the Mach numbers at the duct entrance and exit?
What assumption is made about the kinetic energy change within the chamber in these models?
Why is a discussion of the Rayleigh flow model not offered in the context of filling and evacuating chambers in this chapter?
For a semi-rigid chamber where V(t) = aP^n, what is the value of n for the 'linear condition' if the specific heat ratio k is 1.4?
What happens to the equation for the rapid evacuation of a rigid tank at the transition point to chokeless flow, denoted as chT?
When is the combination of an isentropic process in the chamber and an Isothermal flow in the tube considered a limited case?
For the filling process of a rigid tank with a choked adiabatic isentropic nozzle, if the reduced pressure P_bar is 3.0 after some time, what is the value of P_bar raised to the power of (1-k)/(2k) assuming k=1.4?
In the general equation model (11.13), the term f[M] is introduced as a convenient notation. What does this function represent?
How is the reduced Mach number, M_bar, defined?
What is a key difference between filling/evacuating a chamber through a direct connection versus a connection through a reduction?
For a semi-rigid chamber where V(t) = aP(t)^n, what does the integrated solution for reduced time t_bar look like for the initial part of the evacuation?
According to the model of a simple semi-rigid chamber where V(t)=aP(t), what is the physical meaning of the result that pressure remains larger throughout the evacuating process compared to a rigid tank?
In the derivation of the governing equation for filling a chamber with an adiabatic isentropic nozzle, the initial integration of equation (11.23) leads to equation (11.24). What are the integration limits for the pressure integral?
If a rigid tank is being filled through an isothermal nozzle, the governing differential equation is dP_bar/dt_bar = +/- f[M]*P_bar = 0. What is the reason for this simplified form compared to the general model?
The approximate solution for M_in in the advanced topics section is derived from which two Fanno flow characteristic equations?
A chamber with initial pressure 1.5 Bar is filled from a line until the pressure is 4.5 Bar. What is the final reduced pressure, P_bar?
What is the general governing equation derived in Section 11.2 before any specific process (isentropic, isothermal) is assumed for the chamber?
For a semi-rigid chamber where V(t) = aP(t)^n, and n = (3k-2)/(2k), the relationship between reduced time and pressure becomes linear. What does a value of n above this linear condition cause the pressure-time curve to be?
What is the term for a tank where the volume is either completely rigid or is a function of the chamber's pressure?
If a chamber with volume 0.1 m^3 is filled with air at 10 Bar, connected to a rubber tube with 4fL/D = 50, what flow model should be used for the tube?
In the evacuation of a rigid tank via Fanno flow, the transition from choked to chokeless flow occurs at a point denoted chT. What determines this transition point?
What is the integrated solution for pressure (P_bar) as a function of time (t_bar) for evacuating a rigid tank through an isothermal nozzle, assuming the flow is choked?
In the filling process of a rigid tank with a choked isentropic nozzle, the initial reduced pressure P_bar(0) is 1. After a reduced time t_bar > 0, how will the final P_bar compare to 1?