An air-fuel mixture is compressed by a piston in a cylinder of an internal combu

Question

An air-fuel mixture is compressed by a piston in a cylinder of an internal combustion engine. The origin of coordinate y is at the top of the cylinder, and y points straight down as shown. The piston is assumed to move up at constant speed Vp. The distance L between the top of the cylinder and the piston decreases with time according to the linear approximation L = L bottom - Vpt where L bottom is the location of the piston when it is at the bottom of its cycle at t=0. At t=0. the density of the air-fuel mixture in the cylinder is everywhere equal to p(0). Estimate the density of the air-fuel mixture as a function of time and the given parameters during the piston's up stroke. Density varies with time but not space. Velocity components in the x- and z-directions are zero. Velocity component in the y-direction is a function of y and t. No mass escapes from the cylinder during the compression. hint: Use conservation of mass equation (differential form).

Explanation / Answer

(a) Control Volume ANalysis:

The mass in the cylinder is m = (0)V0 =(0)(r2Lbottom)    (r being the radios of the cylinder)

At time t, the volume becomes    V(t) = r2 (Lbottom-vp t).

Therefore, the density becomes

(t) = m/V(t) = (0)(r2Lbottom) /[r2 (Lbottom-vp t)] = (0)Lbottom / (Lbottom-vp t)

(b) We can also solve this, using differential analysis.

Assume that velocity of the mixture is linearly changed from the piston to the top of the cylinder, i.e.,

v = -y/L(t)*vp = -yvp/(Lbottom - vp t)         (i.e., v = 0 at the top, and v = -vp at the piston)

Therefore, (v)/y = -vp/(Lbottom - vp t)   

The continuity equation is (note the x and z components of velocity u = 0 and w = 0):

/t+(v)/y =0        =>    /t + v/y =0    since is only a function of t.

or   d/dt - vp/(Lbottom - vp t) =0     (change from partial derivative to d/dt is OK since is only a function of t)

=>     d/ = dt vp/(Lbottom - vp t)      =>

ln = -ln(Lbottom - vp t)+ lnC = ln[C/( Lbottom - vp t)]                      (C is a constant)     =>

(t) = C/( Lbottom - vp t)

Use initial condition (0)=C/Lbottom, we have C = (0)Lbottom, and

  (t) = (0)Lbottom/( Lbottom - vp t)   which is the same as the result from control volume analysis.

              

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