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General Model and Non-dimensioned

It is convenient to non-dimensioned the properties in chamber by dividing them by their initial conditions. The dimensionless properties of chamber as

$\displaystyle \bar{T} = { T(t=\bar{t}) \over T(t=0) }$

(11.4)

$\displaystyle \bar{V} = { V(t=\bar{t}) \over V(t=0) }$

(11.5)

$\displaystyle \bar{P} = { P(t=\bar{t}) \over P(t=0) }$

(11.6)

$\displaystyle \bar{t} = { t \over t_c }$

(11.7)

where

is the characteristic time of the system defined as followed

$\displaystyle t_c = { V(0) \over A M_{max} \sqrt{kRT(0)})}$

(11.8)

The physical meaning of characteristic time,

is the time that will take to evacuate the chamber if the gas in the chamber was in its initial state, the flow rate was at its maximum (choking flow), and the gas was incompressible in the chamber.

Utilizing these definitions (11.4) and substituting into equation (11.3) yields

$\displaystyle { P(0) V(0) \over t_c R T(0) } {d \over d\bar{t}}{\left( {\bar{P}... ...ho} A {\overbrace{\sqrt{kR\bar{T}_1T(0)}}^{c(t)} } M_{max} \bar{M}(\bar{t}) = 0$

(11.9)

where the following definition for the reduced Mach number is added as

$\displaystyle \bar{M} = {M_1(t) \over M_{max}}$

(11.10)

After some rearranging equation (11.6) obtains the form

$\displaystyle {d \over d \bar{t}}{\left( {\bar{P} \bar{V} \over \bar{T}}\right)... ...RT(0)}} \over V(0)} {{\bar{P_1} \bar{M}_1 \over \sqrt{\bar{T_1}}} } \bar{M} = 0$

(11.11)

and utilizing the definition of characteristic time, equation (11.5), and substituting into equation (11.8) yields

$\displaystyle {d \over d \bar{t}}{\left( {\bar{P} \bar{V} \over \bar{T}}\right)} \pm {{\bar{P_1} \bar{M} \over \sqrt{\bar{T_1}}} } = 0$

(11.12)

Note that equation (11.9) can be modified by introducing additional parameter which referred to as external time, $t_{max}$ ^11.3. For cases, where the process time is important parameter equation (11.9) transformed to

$\displaystyle {d \over d \tilde{t}} {\left( {\bar{P} \bar{V} \over \bar{T}} \right) } \pm {t_{max} \over t_{c}} {\bar{P_1} \bar{M} \over \sqrt{\bar{T_1}}} = 0$

(11.13)

when $\bar{P}, \bar{V},\bar{T},$ and $\bar{M}$ are all are function of $\tilde{t}$ in this case. And where $\tilde{t} = t / t_{max}$ .

It is more convenient to deal with the stagnation pressure then the actual pressure at the entrance to the tube. Utilizing the equations developed in Chapter 4 between the stagnation condition, denoted without subscript, and condition in a tube denoted with subscript 1. The ratio of $\bar{P_1} \over \sqrt{\bar{T_1}}$ is substituted by

$\displaystyle {\bar{P_1} \over \sqrt{\bar{T_1}}} = {\bar{P} \over \sqrt{\bar{T}}} \left[{ 1 + {k-1 \over 2} M^{2} } \right] ^{-(k+1) \over 2(k-1)}$

(11.14)

It is convenient to denote

$\displaystyle f[ M ] = \left[ 1 + {k-1 \over 2} {M}^{2} \right] ^ {\frac{-(k+1)}{2(k-1)}}$

(11.15)

Note that

is a function of the time. Utilizing the definitions (11.11) and substituting equation (11.12) into equation (11.9) to be transformed into

$\displaystyle {d \over d \bar{t}} {\left( {\bar{P} \bar{V} \over \bar{T}} \right) } \pm {{\bar{P} \bar{M}(\bar{t}) f[M] \over \sqrt{\bar{T}}} } = 0$

(11.16)

Equation (11.13) is a first order nonlinear differential equation that can be solved for different initial conditions. At this stage, the author isn't aware that there is a general solution for this equation^11.4. Nevertheless, many numerical methods are available to solve this equation.

Subsections

Next: Isentropic Process Up: Evacuating SemiRigid Chambers Previous: Governing Equations and Assumptions Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21

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