Question

A liquid-vapor mixture of refrigerant-134a is at \(280 \mathrm{kPa}\) with a quality of 70 percent. Determine the value of the Gibbs function, in \(\mathrm{kJ} / \mathrm{kg},\) when the two phases are in equilibrium.

Short answer

Question: Calculate the Gibbs function for a liquid-vapor mixture of refrigerant-134a at 280 kPa and having a quality of 70%. Answer: To find the Gibbs function of liquid-vapor mixture, follow these steps: 1. Look up the specific enthalpy and specific entropy values of the saturated liquid and saturated vapor for refrigerant-134a at 280 kPa in a thermodynamic properties table. Denote these values as \(h_f\), \(h_g\), \(s_f\), and \(s_g\). 2. Calculate the specific enthalpy and specific entropy of the mixture using the formula: \(h_{mixture} = h_f + 0.7(h_g - h_f)\) and \(s_{mixture} = s_f + 0.7(s_g - s_f)\), since the quality is 70%. 3. Look up the saturation temperature at 280 kPa in a refrigerant-134a table or thermodynamic properties table, and convert it to Kelvin. 4. Calculate the Gibbs function using the formula: \(G = h_{mixture} - T \times s_{mixture}\), where T is the saturation temperature in Kelvin, and \(h_{mixture}\) and \(s_{mixture}\) are the values obtained in step 2. The resulting Gibbs function value will represent the equilibrium state of the refrigerant-134a liquid-vapor mixture at 280 kPa and 70% quality.

Step-by-step solution

Determine the specific enthalpy and specific entropy values of the saturated liquid and saturated vapor

First, we need to look up the specific enthalpy and specific entropy values of the saturated liquid and saturated vapor for refrigerant-134a at the given pressure (280 kPa). You can find this data in a refrigerant-134a table or a thermodynamic properties table. Let's denote the specific enthalpy values for the saturated liquid and the saturated vapor as \(h_f\) and \(h_g\) respectively, and the specific entropy values for the saturated liquid and saturated vapor as \(s_f\) and \(s_g\) respectively.

Calculate the specific enthalpy and specific entropy of the mixture using the quality

We know that the quality of the mixture is 70%, which means that 70% of the substance exists as a vapor phase and 30% as a liquid phase. We can use the quality to find the specific enthalpy and specific entropy of the mixture. The specific enthalpy (\(h_{mixture}\)) can be calculated using the formula: \(h_{mixture} = h_f + x(h_g - h_f)\), where x is the quality. Similarly, the specific entropy (\(s_{mixture}\)) can be calculated using the formula: \(s_{mixture} = s_f + x(s_g - s_f)\).

Calculate the Gibbs function using the specific enthalpy and entropy values

Now that we have the specific enthalpy and specific entropy values for the mixture, we can use the Gibbs function definition to find the value of the Gibbs function (\(G\)) for the refrigerant-134a mixture. The relation between the Gibbs function, specific enthalpy, specific entropy, and temperature is given by: \(G = h_{mixture} - T \times s_{mixture}\) To calculate the Gibbs function in kJ/kg, substitute the specific enthalpy, specific entropy, and temperature (in Kelvin) into the formula. Note that the given pressure was 280 kPa, so you will need to look up the saturation temperature at this pressure in a refrigerant-134a table or thermodynamic properties table. With all the values substituted in the formula, calculate the Gibbs function for the refrigerant-134a liquid-vapor mixture when the two phases are in equilibrium.