A cylinder fitted with a piston is filled with 700 lb of saturated liquid ammonia at 15°F. The piston weighs 1 ton and has a diameter of 2.5 ft. Given Patm = 1 atm, what is the vol... A cylinder fitted with a piston is filled with 700 lb of saturated liquid ammonia at 15°F. The piston weighs 1 ton and has a diameter of 2.5 ft. Given Patm = 1 atm, what is the volume occupied by the ammonia, in ft³? Ignoring friction, determine the force required, in lbf, by mechanical attachments to hold the piston in place. Read more

Steps to Solve

1. Find the specific volume of saturated liquid ammonia at 15°F

   From thermodynamics tables (or software), the specific volume ($v_f$) of saturated liquid ammonia at 15°F is approximately 0.02555 $ft^3/lb$.

2. Calculate the volume of the ammonia

   Multiply the specific volume by the mass of the ammonia to find the total volume: $V = m \cdot v_f$ $V = 700 , lb \cdot 0.02555 , ft^3/lb = 17.885 , ft^3$

3. Calculate the pressure inside the cylinder

   The pressure inside the cylinder is due to the atmospheric pressure plus the pressure exerted by the piston's weight:

   $P_{internal} = P_{atm} + P_{piston}$

   First, calculate the area of the piston: $A = \pi r^2 = \pi (\frac{D}{2})^2 = \pi (\frac{2.5 , ft}{2})^2 = \pi (1.25 , ft)^2 \approx 4.9087 , ft^2$

   Convert the piston's weight from tons to lbf (1 ton = 2000 lb): $Weight_{piston} = 1 , ton \cdot 2000 , lb/ton = 2000 , lb$

   Calculate the pressure exerted by the piston: $P_{piston} = \frac{Weight_{piston}}{A} = \frac{2000 , lb}{4.9087 , ft^2} \approx 407.43 , lb/ft^2$

   Convert atmospheric pressure from atm to lb/ft² (1 atm = 14.696 psi = 2116.224 lb/ft²): $P_{atm} = 1 , atm \cdot 2116.224 , lb/ft^2/atm = 2116.224 , lb/ft^2$

   Now, calculate total internal pressure: $P_{internal} = P_{atm} + P_{piston} = 2116.224 , lb/ft^2 + 407.43 , lb/ft^2 = 2523.654 , lb/ft^2$

4. Determine the force required to hold the piston in place

   If the force is negative this means the attachments are holding piston down If the force is positive this means the attachments are holding the piston up

   The force balance on the piston is: $F_{internal} = F_{atm} + F_{piston} + F_{attachments}$ $P_{internal} \cdot A = P_{atm} \cdot A + Weight_{piston} + F_{attachments}$

   Therefore: $F_{attachments} = (P_{internal} - P_{atm}) \cdot A - Weight_{piston}$ $F_{attachments} = P_{piston} \cdot A - Weight_{piston}$ $F_{attachments} = 407.43 , lb/ft^2 \cdot 4.9087 , ft^2 - 2000 , lb$ $F_{attachments} = 2000 , lb - 2000 , lb = 0 , lb$