Finding Thermodynamic Properties

Finding Thermodynamic Properties These notes outline a process for finding the thermodynamic state when you are given two thermodynamic properties. The first two topics consider the cases where you are given any two of the following three properties: temperature, pressure, and specific volume. Following that, we consider the use of other thermodynamic properties to find the state. In finding the state, you will first have to determine if you have a compressed liquid, a gas, or a mixture of liquid and vapor A. If you are given temperature and pressure, you have the following possibilities. 1. It will not be in the mixed region since temperature and pressure are not independent in this region. 2. The substance is in the liquid state if the pressure is greater than the saturation pressure at the given temperature or if the temperature is less than the saturation temperature at the given pressure. In this case, you may have compressed liquid tables for high pressures or else you can estimate the properties as those of the saturated liquid at the given temperature. 3. The substance is in the gas (superheated vapor) state if the pressure is less than the saturation pressure at the given temperature or if the temperature is greater than the saturation temperature at the given pressure. Here you will use the gas phase (superheat) tables, which have pressure and temperature as independent lookup variables to find the properties. 4. Example: Water with T = 100oC and P = 200 kPa. We see that Psat(100oC) = 101.35 kPa which is less than the given P. Thus we are in a compressed liquid state. (Alternatively we could have found Tsat(200 kPa) = 120.23oC. Since this is less than the given temperature of 100oC, we conclude that we have a compressed liquid.) We use the compressed liquid properties at 100oC to estimate the properties at 100oC and 200 kPa; these are v = 0.001044 m3/kg, u = 418.94 kJ/kg, h = 419.04 kJ/kg and s = 1.3069 kJ/kg∙K. 5. Another example: Water with T = 200oC and 100 kPa. We find this state in the superheat tables with the following properties: v = 2.172 m3/kg, u = 2658.1 kJ/kg, h = 2875.3 kJ/kg and s = 7.8343 kJ/kg∙K. Here we went straight to the superheat tables. If we find the state there we know that it is in the gas phase. Note that the saturation temperature at 100 kPa, which is 99.63oC, is shown next to the pressure on the superheat table. This confirms that the saturation temperature is less than the given temperature so that we are in the gas phase. B. If you are given temperature and specific volume (or pressure and specific volume), use the following process. 1. Use the given pressure or the given temperature to find the specific volumes of the saturated liquid (vf) and saturated vapor (vg). Page 2 2. If the given specific volume is less than the specific volume of the saturated liquid, you are in the compressed liquid state. In this case, you may have compressed liquid tables for high pressures or else you can estimate the properties as those of the saturated liquid at the given temperature. (If you are given the pressure, you have to find the saturation temperature to make the approximation for the compressed liquid.) 3. If the given specific volume is greater than the specific volume of the saturated vapor, you are in the superheated vapor (gas) state. Here you use the given temperature (or pressure) and specific volume in the gas tables. This usually requires an interpolation to find the given specific volume. 4. If the given specific volume is greater than the specific volume of the saturated liquid and less than the specific volume of the saturated vapor, you are in the mixed region. Here you find the quality as (v - vf)/(vg - vf). You can then use this quality to find other properties, such as the internal energy, as u = uf + x(ug - uf). 5. Example. For water at 200 kPa with a specific volume of 0.5 m3/kg we find that the specific volumes of the saturated liquid and saturated vapor at the given pressure of 200 kPa are vf = 0.001061 m3/kg and vg = 0.8857 m3/kg. Thus we are in the mixed region and we have to compute the quality as x = (v – vf)/(vg – vf) = (0.5 – 0.001061) / (0.8857 – 0.001061) = 0.564. Once we know the quality we can find all other properties. For example the enthalpy is found as h = hf + x hfg = 504.70 + (0.564)(2201.9) = 1746.6 kJ/kg. 6. Another example. For water at 500oF with a specific volume of 6 ft3/lbm, we see that this specific volume is greater than the specific volume of the saturated liquid at 500oF, which is 0.6761 m3/kg so we know that we are in the gas (superheat) region. Scanning the tables for T = 500oF, we find a specific volume of 7.017 ft3/lbm at 80 psia and a specific volume of 5.587 ft3/lbm at 100 psia. Thus our given specific volume of 6 ft3/lbm must be between these pressures. We have to interpolate using the following equation for a general property Y which may equal P, u, s, or h. First we find the interpolation factor, f, which is a measure of the relative distance between the two data points. Note that f will be negative in this case because v is decreasing as P is increasing. f = 3 6 ft − 7.017 ft 3 lbm lbm 3 3 5.587 ft − 7.017 ft lbm lbm We then use this factor to compute any property that is tabulated at the two pressures. In the equation below, Y represents u, h, s, or P. 3  = Y (500 0 F ,80 psia ) + f (Y (500 0 F ,100 psia ) − Y (500 0 F ,80 psia ) ) Y  500 0 F ,6 ft  lb m   C. If you are given temperature (or pressure) and another specific property (specific internal energy, specific enthalpy, or specific entropy), follow the same process used as in part B. This is written below using entropy as a specific example, but you could use energy or enthalpy. 1. Use the given pressure or the given temperature to find the specific entropy of the saturated liquid (sf) and saturated vapor (sg). Page 3 2. If the given specific entropy is less than the specific entropy of the saturated liquid, you are in the compressed liquid state. In this case, you may have compressed liquid tables for high pressures or else you can estimate the properties as those of the saturated liquid at the given temperature. (If you are given the pressure, you have to find the saturation temperature to make the approximation for the compressed liquid.) 3. If the given specific entropy is greater than the specific entropy of the saturated vapor, you are in the superheated vapor (gas) state. Here you use the given temperature (or pressure) and specific entropy in the gas tables. This usually requires an interpolation to find the given specific entropy. 4. If the given specific entropy is greater than the specific entropy of the saturated liquid and less than the specific entropy of the saturated vapor, you are in the mixed region. Here you find the quality as (s - sf)/(sg - sf). You can then use this quality to find other properties, such as the internal energy, as u = uf + x(ug - uf). 5. The examples in this case proceed in exactly the same way as the examples for case B. Only the other specified property replaces the specific volume in the examples.