Rapid Process

Clearly under the assumption of rapid process the heat transfer can be neglected and Fanno flow can be assumed for the tube. The first approximation isotropic process describe the process inside the cylinder (see Figure (12.1)).

Figure 12.1: The control volume of the ``Cylinder''.

Before introducing the steps of the analysis, it is noteworthy to think about the process in qualitative terms. The replacing incompressible liquid enter in the same amount as replaced incompressible liquid. But in a compressible substance the situation can be totally different, it is possible to obtain a situation where that most of the liquid entered the chamber and yet most of the replaced gas can be still be in the chamber. Obtaining conditions where the volume of displacing liquid is equal to the displaced liquid are called the critical conditions. These critical conditions are very significant that they provide guidelines for the design of processes.

Obviously, the best ventilation is achieved with a large tube or area. In manufacture processes to minimize cost and the secondary machining such as trimming and other issues the exit area or tube has to be narrow as possible. In the exhaust system cost of large exhaust valve increase with the size and in addition reduces the strength with the size of valve^12.2. For these reasons the optimum size is desired. The conflicting requirements suggest an optimum area, which is also indicated by experimental studies and utilized by practiced engineers.

The purpose of this analysis to yields a formula for critical/optimum vent area in a simple form is one of the objectives of this section. The second objective is to provide a tool to ``combine'' the actual tube with the resistance in the tube, thus, eliminating the need for calculations of the gas flow in the tube to minimize the numerical calculations.

A linear function is the simplest model that decibels changes the volume. In reality, in some situations like die casting this description is appropriate. Nevertheless, this model can be extended numerical in cases where more complex function is applied.

$$V(t) = V(0) \left[ 1 - {t \over t_{max}} \right]$$ (12.1)

Equation (12.1) can be non-dimensionlassed as

$$\bar{V}(\bar{t}) = 1 - \bar{t}$$ (12.2)

The governing equation (11.10) that was developed in the previous Chapter (11) obtained the form as

$$\left[{\bar{P}} \right] ^{ 1 \over k} \left\{ {1 \over k} {\bar{V... ...} + {t_{max} \bar{M} f(M) \over t_c} \left[{\bar{P}}\right] ^{k+1 \over 2k} = 0$$ (12.3)

where $\bar{t} = t/t_{max}$ . Notice that in this case that there are two different characteristic times: the ``characteristic'' time, $t_c$ and the ``maximum'' time, $t_{max}$ . The first characteristic time, $t_c$ is associated with the ratio of the volume and the tube characteristics (see equation (11.5)). The second characteristic time, $t_{max}$ is associated with the imposed time on the system (in this case the elapsed time of the piston stroke).

Equation (12.3) is an nonlinear first order differential equation and can be rearranged as follows

$${d\bar{P} \over k \left(1- {t_{max} \over t_c}{\bar{M}}f[M] \bar{... ...P}} = {d\bar{t} \over 1-\bar{t}} \qquad ; \;\;\;\;\;\;\;\;\;\;\;\bar{P}(0) = 1.$$ (12.4)

Equation (12.4) is can be solved only when the flow is chocked In which case $f[m]$ isn't function of the time.

The solution of equation (12.4)) can be obtained by transforming and by introducing a new variable $\xi = \bar{P}^{k-1 \over 2k}$ and therefore $\bar{P} = \left[ \xi \right]^{2k \over k -1}$ . The reduced Pressure derivative, $d\bar{P} = {2k \over k -1} \left[ \xi \right]^{\left(2k \over k -1\right) -1} d\xi$ Utilizing this definition and there implication reduce equation (12.4)

$${2 \left[ \xi \right]^{\left(2k \over k -1\right) -1} d\xi \over ... ... B \xi \right) \left[ \xi \right]^{2k \over k -1}} = {d\bar{t} \over 1-\bar{t}}$$ (12.5)

where $B = {t_{max} \over t_c}{\bar{M}}f[M]$ And equation (12.5) can be further simplified as

$${2 d\xi \over (k -1) \left(1- B \xi \right) \xi } = {d\bar{t} \over 1-\bar{t}}$$ (12.6)

Equation (12.6) can be integrated to obtain

$${2 \over (k-1) B} \ln \left\vert 1 -B \xi \over \xi \right\vert = - \ln \bar{t}$$ (12.7)

or in a different form

$${\left\vert 1 -B \xi \over \xi \right\vert}^{2 \over (1-k) B} = \bar{t}$$ (12.8)

Now substituting to the ``preferred'' variable

$$\left. {\left[ 1 - {t_{max} \over t_c}{\bar{M}}f[M] \bar{P}^{k-1 ... ...\over (1-k) {t_{max} \over t_c}{\bar{M}}f[M]} \right\vert^1_{\bar{P}} = \bar{t}$$ (12.9)

The analytical solution is applicable only in the case which the flow is choked thorough all the process. The solution is applicable to indirect connection. This happen when vacuum is applied outside the tube (a technique used in die casting and injection molding to improve quality by reducing porosity.). In case when the flow chokeless a numerical integration needed to be performed. In the literature, to create a direct function equation (12.4) is transformed into

$${d \bar{P} \over d \bar{t}} = {k\left(1 - {t_{max} \over t_c}{\bar{M}}f[M] \bar{P}^{k-1 \over 2k} \right) \over 1 - \bar{t}}$$ (12.10)

with the initial condition of

$$P(0) = 1$$ (12.11)

Figure 12.2: The pressure ratio as a function of the dimensionless time for chokeless condition

The analytical solution also can be approximated by a simpler equation as

$$\bar{P} = \left[ 1 -t\right] ^ {t_{max} \over t_c}$$ (12.12)

The results for numerical evaluation in the case when cylinder is initially at an atmospheric pressure and outside tube is also at atmospheric pressure are presented in Figure (12.2). In this case only some part of the flow is choked (the later part). The results of a choked case are presented in Figure (12.3) in which outside tube condition is in vacuum. These Figures (12.2) and 12.3 demonstrate the importance of the ratio of ${t_{max} \over t_c}$ . When ${t_{max} \over t_c} > 1$ the pressure increases significantly and verse versa.

Figure: $\bar{P}$ as a function of $\bar{t}$ for choked condition

Thus, the question remains how the time ratio can be transfered to parameters that can the engineer can design in the system.

Denoting the area that creates the ratio ${t_{max} \over t_c} = 1$ as the critical area, $A_c$ provides the needed tool. Thus the exit area, $A$ can be expressed as

$$A = {A \over A_c} A_c$$ (12.13)

The actual times ratio $\left. {t_{max} \over t_c}\right\vert _{@A}$ can be expressed as

$$\left. {t_{max} \over t_{c}} \right\vert _{@A} = \left. {t_{max} ... ...ght\vert _{@A} \overbrace{\left. {t_{max} \over t_{c}} \right\vert _{@A_c}}^{1}$$ (12.14)

According to equation (11.5) $t_c$ is inversely proportional to area, $t_c \propto 1/A$ . Thus, equation (12.14) the $t_{max}$ is canceled and reduced into

$$\left. {t_{max} \over t_{c}} \right\vert _{@A} = {A \over A_c}$$ (12.15)

Parameters influencing the process are the area ratio, $A \over A_c$ , and the friction parameter, $\frac{4fL}{D}$ . From other detailed calculations the author thesis (later to be published on this www.potto.org). it was found that the influence of the parameter $\frac{4fL}{D}$ on the pressure development in the cylinder is quite small. The influence is small on the residual air mass in the cylinder but larger on the Mach number, $M_{exit}$ . The effects of the area ratio, $A \over A_c$ , are studied here since it is the dominant parameter.

It is important to point out the significance of the $t_{max} \over t_{c}$ . This parameter represents the ratio between the filling time and the evacuating time, the time which would be required to evacuate the cylinder for constant mass flow rate at the maximum Mach number when the gas temperature and pressure remain in their initial values. This parameter also represents the dimensionless area, $A \over A_c$ , according to the following equation

Figure (12.4) describes the pressure as a function of the dimensionless time for various values of $A \over A_c$ . The line that represents ${A \over A_c} =1$ is almost straight.

Figure 12.4: The pressure ratio as a function of the dimensionless time

For large values of $A \over A_c$ the pressure increases the volume flow rate of the air until a quasi steady state is reached. This quasi steady state is achieved when the volumetric air flow rate out is equal to the volume pushed by the piston. The pressure and the mass flow rate are maintained constant after this state is reached. The pressure in this quasi steady state is a function of $A \over A_c$ . For small values of $A \over A_c$ there is no steady state stage. When $A \over A_c$ is greater than one the pressure is concave upward and when $A \over A_c$ is less than one the pressure is concave downward as shown in Figures (12.4), which was obtained by an integration of equation (12.9).