2. An air -fuel mixture is compressed by a piston in a cylinder of an internal c
ID: 1767151 • Letter: 2
Question
2. An air -fuel mixture is compressed by a piston in a cylinder of an internal combustion engine. The piston is assumed to move up at constant s peed VP. The distance L between the top of the cylinder and the piston decreases linearly with time. The density of the air-fuel mixture was calculated in function of time and it is equal to p o etaAssume there is a two dimensional flow, where the velocity in the y-direction varies linearly with y but not with time, al to p Po Lpottom-Vip but the u-velocity in the xdirection has a more complex variation. Assuming that the piston (10 points) (20 points width is b, determine: The linear variation of the v velocotiy (y-direction) The u-velocity component using the mass conservation equation a. b. Cylinder LI L(O V Piston pC) bottom Time t Timet 0 VpExplanation / 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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