Water contained in a piston-cylinder assembly, initially at 300°F, a quality of 80%, and a volume of 6 ft^3, is heated at a constant temperature to saturated vapor. If the rate of... Water contained in a piston-cylinder assembly, initially at 300°F, a quality of 80%, and a volume of 6 ft^3, is heated at a constant temperature to saturated vapor. If the rate of heat transfer is 0.4 Btu/s, determine the time, in minutes, for this process of the water to occur. Kinetic and potential energy effects are negligible.
Understand the Problem
The question describes a thermodynamic process involving water in a piston-cylinder assembly. The water is initially at a given temperature, quality, and volume, and it's heated at constant temperature until it becomes saturated vapor. Given a rate of heat transfer, the question asks to calculate the time required for the water to reach saturated vapor, while neglecting kinetic and potential energy effects.
Answer
$t = 8.78 \ min$
Answer for screen readers
$t = 8.78 \ min$
Steps to Solve
- Find the specific volume at the initial state
The specific volume $v_1$ at the initial state can be calculated using the given quality $x_1 = 0.8$ and the specific volumes of saturated liquid $v_f$ and saturated vapor $v_g$ at 300°F. From saturated water tables: $v_f = 0.01745 \ ft^3/lb$ and $v_g = 6.364 \ ft^3/lb$.
The specific volume at the initial state is calculated as follows: $$v_1 = v_f + x_1(v_g - v_f) = 0.01745 + 0.8(6.364 - 0.01745) = 5.0947 \ ft^3/lb$$
- Calculate the mass of the water
The mass $m$ of the water can be found using the initial volume $V_1 = 6 \ ft^3$ and the specific volume $v_1$ : $$m = \frac{V_1}{v_1} = \frac{6}{5.0947} = 1.1777 \ lb$$
- Find the specific enthalpy at the initial and final states
The specific enthalpy at initial state $h_1$ can be calculated using the given quality $x_1 = 0.8$ and the specific enthalpies of saturated liquid $h_f$ and saturated vapor $h_g$ at 300°F. From saturated water tables: $h_f = 269.73 \ Btu/lb$ and $h_g = 1164.1 \ Btu/lb$.
The specific enthalpy at the initial state is calculated as follows: $$h_1 = h_f + x_1(h_g - h_f) = 269.73 + 0.8(1164.1 - 269.73) = 985.23 \ Btu/lb$$
At the final state, the water is saturated vapor at 300°F, so $h_2 = h_g = 1164.1 \ Btu/lb.$
- Apply the energy balance equation
Since kinetic and potential energy effects are negligible, the energy balance equation for a constant-pressure process simplifies to: $Q = m(h_2 - h_1)$ Where $Q$ is the total heat transfer. $$Q = 1.1777(1164.1 - 985.23) = 210.66 \ Btu$$
- Calculate the time required for the process
Given the rate of heat transfer $\dot{Q} = 0.4 \ Btu/s$, the time $t$ required for the process can be calculated as: $$t = \frac{Q}{\dot{Q}} = \frac{210.66}{0.4} = 526.65 \ s$$
- Convert the time to minutes
Convert the time from seconds to minutes: $$t = \frac{526.65}{60} = 8.78 \ min$$
$t = 8.78 \ min$
More Information
The problem involves a constant-pressure phase change (heating water to saturated vapor). The key is to use steam tables to find the specific enthalpy values at different states, and then apply the energy balance equation to solve for the heat transfer. Finally, dividing the total heat transfer by the rate of heat transfer gives the time required for the process.
Tips
A common mistake is using the wrong steam table values or not using interpolation when necessary. Also, one could forget to convert the final answer to the requested units (minutes). Another common mistake is neglecting the change is specific volume in calculating the mass of the water.
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