As shown in the figure below, 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. What... As shown in the figure below, 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. What is the volume occupied by the ammonia, in ft³? Ignoring friction, determine the force required, in lbf, by mechanical attachments, such as stops, to hold the piston in place. Read more

Steps to Solve

1. Find the specific volume of saturated liquid ammonia

   From thermodynamics tables, the specific volume $v_f$ of saturated liquid ammonia at 15°F is approximately $0.02555 \text{ ft}^3/\text{lb}$.

2. Calculate the total volume of the ammonia

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

3. Calculate the area of the piston

   The diameter of the piston is given as 2.5 ft. Therefore, the radius $r$ is 1.25 ft. Calculate the area $A$ of the piston:

   $$ A = \pi r^2 = \pi (1.25 \text{ ft})^2 \approx 4.9087 \text{ ft}^2 $$

4. Calculate the force due to atmospheric pressure

   The atmospheric pressure is given as 1 atm, which is approximately $14.696 \text{ lb}/\text{in}^2$. Convert this to $\text{lb}/\text{ft}^2$: $$ P_{atm} = 14.696 \frac{\text{lb}}{\text{in}^2} \cdot \frac{144 \text{ in}^2}{1 \text{ ft}^2} = 2116.224 \frac{\text{lb}}{\text{ft}^2} $$ Calculate the force due to atmospheric pressure: $$ F_{atm} = P_{atm} \cdot A = 2116.224 \frac{\text{lb}}{\text{ft}^2} \cdot 4.9087 \text{ ft}^2 \approx 10387.22 \text{ lb} $$

5. Calculate the weight of the piston

   The piston weighs 1 ton, which is equal to 2000 lb. Thus, the force due to the weight of the piston is: $$ F_{piston} = 2000 \text{ lb} $$

6. Determine the absolute pressure inside the cylinder

   The absolute pressure inside the cylinder is the saturation pressure at 15°F. From thermodynamics tables, this is $37.23 \text{ psia}$

7. Convert the internal pressure to $lb/ft^2$ $$ P_{internal} = 37.23 \frac{\text{lb}}{\text{in}^2} \cdot \frac{144 \text{ in}^2}{1 \text{ ft}^2} = 5361.12 \frac{\text{lb}}{\text{ft}^2} $$

8. Calculate the upward force exerted by the ammonia

   The upward force exerted by the ammonia is the internal pressure times the area of the piston.

   $$ F_{up} = P_{internal} \cdot A = 5361.12 \frac{\text{lb}}{\text{ft}^2} \cdot 4.9087 \text{ ft}^2 \approx 26314.03 \text{ lb} $$

9. Calculate the net upward force

   The net upward force from the ammonia is the upward force minus the atmospheric force minus the piston weight. $$F_{net_up} = F_{up} - F_{atm} - F_{piston} = 26314.03 \text{ lb} - 10387.22 \text{ lb} - 2000 \text{ lb} = 13926.81 \text{ lb}$$

10. Determine the force required by mechanical attachments

    Since the net force is upward, the mechanical attachments must exert a downward force to hold the piston in place. Therefore: $$F_{attachments} = F_{net_up} = 13926.81 \text{ lb}$$