Question

Using the tables for water, determine the specified property data at the indicated states. Check the results using $IT$. In each case, locate the state by hand on sketches of the $p-v$ and $T-v$ diagrams. (a) At $p=3$ bar, $T={240}^{\circ }\mathrm{C}$, find $v$ in ${\mathrm{m}}^3/\mathrm{kg}$ and $u$ in $\mathrm{kJ}/\mathrm{kg}$. (b) At $p=3$ bar, $v=0.5{\mathrm{m}}^3/\mathrm{kg}$, find $T$ in ${}^{\circ }\mathrm{C}$ and $u$ in $\mathrm{kJ}/\mathrm{kg}$. (c) At $T={400}^{\circ }\mathrm{C},p=10$ bar, find $v$ in ${\mathrm{m}}^3/\mathrm{kg}$ and $h$ in $\mathrm{kJ}/\mathrm{kg}$. (d) At $T={320}^{\circ }\mathrm{C},v=0.03{\mathrm{m}}^3/\mathrm{kg}$, find $p$ in $\mathrm{MPa}$ and $u$ in $\mathrm{kJ}/\mathrm{kg}$ (e) At $p=28\mathrm{MPa},T={520}^{\circ }\mathrm{C}$, find $v$ in ${\mathrm{m}}^3/\mathrm{kg}$ and $h$ in $\mathrm{kJ}/\mathrm{kg}$. (f) At $T={100}^{\circ }\mathrm{C},x=60\mathrm{\%}$, find $p$ in bar and $v$ in ${\mathrm{m}}^3/\mathrm{kg}$. (g) At $T={10}^{\circ }\mathrm{C},v=100{\mathrm{m}}^3/\mathrm{kg}$, find $p$ in $\mathrm{kPa}$ and $h$ in $\mathrm{kJ}/\mathrm{kg}$. (h) At $p=4\mathrm{MPa},T={160}^{\circ }\mathrm{C}$, find $v$ in ${\mathrm{m}}^3/\mathrm{kg}$ and $u$ in $\mathrm{kJ}/\mathrm{kg}$.

Short answer

Use steam tables to locate property data for each state based on provided conditions and sketch the state in the relevant diagrams.

Step-by-step solution

Find specific volume and internal energy at given pressure and temperature

Given: \ Pressure, $p=3\text{ bar}=300\text{ kPa}$ \ Temperature, $T={240}^{\circ }\text{C}$ \ Using water property tables, locate the data for the given conditions. If the conditions fall under the superheated region, use the superheated steam tables to find specific volume ($v$) and internal energy ($u$).

Verify the state and locate on diagrams

Check the pressure and temperature against the saturation properties. Since $T>T_{sat}$, the state is superheated. Sketch the state on the $p-v$ and $T-v$ diagrams.

Find temperature and internal energy at given pressure and specific volume

Given: \ Pressure, $p=3\text{ bar}=300\text{ kPa}$ \ Specific Volume, $v=0.5{\text{ m}}^3/\text{kg}$ \ Use the water property tables to find the corresponding temperature ($T$) and internal energy ($u$).

Verify the state and locate on diagrams

Compare the given $v$ with values in the tables. If $v_f<v<v_g$, the state is in the vapor dome (mixture). If $v>v_g$, the state is superheated. Sketch the state on the $p-v$ and $T-v$ diagrams.

Determine specific volume and enthalpy at given conditions

Given: \ Temperature, $T={400}^{\circ }\text{C}$ \ Pressure, $p=10\text{ bar}=1000\text{ kPa}$ \ Use the superheated steam tables to find specific volume ($v$) and enthalpy ($h$).

Verify the state and locate on diagrams

Sketch the state on the $p-v$ and $T-v$ diagrams after verifying that the conditions are in the superheated region.

Find pressure and internal energy at given temperature and specific volume

Given: \ Temperature, $T={320}^{\circ }\text{C}$ \ Specific Volume, $v=0.03{\text{ m}}^3/\text{kg}$ \ Use the water property tables to find the corresponding pressure ($p$) and internal energy ($u$).

Verify the state and locate on diagrams

Compare $v$ with the saturation values. If $v_f<v<v_g$, the state is a mixture. If $v>v_g$, the state is superheated. Sketch the state on the $p-v$ and $T-v$ diagrams.

Determine specific volume and enthalpy at given pressure and temperature

Given: \ Pressure, $p=28\text{ MPa}$ \ Temperature, $T={520}^{\circ }\text{C}$ \ Use the supercritical steam tables to find specific volume ($v$) and enthalpy ($h$).

Verify the state and locate on diagrams

Sketch the state on the $p-v$ and $T-v$ diagrams after confirming the supercritical conditions.

Find pressure and specific volume at given temperature and dryness fraction

Given: \ Temperature, $T={100}^{\circ }\text{C}$ \ Quality, $x=0.6$ \ Use the saturation tables to find the corresponding pressure ($p$) and specific volumes ($v_f$ and $v_{fg}$). Calculate specific volume ($v$) using $v=v_f+x{v}_{fg}$.

Verify the state and locate on diagrams

Sketch the state (saturated mixture) on the $p-v$ and $T-v$ diagrams.

Determine pressure and enthalpy at given temperature and specific volume

Given: \ Temperature, $T={10}^{\circ }\text{C}$ \ Specific Volume, $v=100{\text{ m}}^3/\text{kg}$ \ Use the water property tables to find the corresponding pressure ($p$) and enthalpy ($h$).

Verify the state and locate on diagrams

Identify the region comparing $v$ with the table values. Sketch the state on the $p-v$ and $T-v$ diagrams.

Find specific volume and internal energy at given pressure and temperature

Given: \ Pressure, $p=4\text{ MPa}$ \ Temperature, $T={160}^{\circ }\text{C}$ \ Use the water property tables to find specific volume ($v$) and internal energy ($u$).

Verify the state and locate on diagrams

Sketch the state on the $p-v$ and $T-v$ diagrams after verifying the conditions specified.

Key concepts

Specific volume in Thermodynamics

Specific volume is a key property in thermodynamics that represents the volume occupied by a unit mass of a substance. In simpler terms, it is the reciprocal of density. Specific volume ($v$) is given by the formula: $$v=\frac{V}{m}$$Where $V$ is the volume and $m$ is the mass. Think of it this way: if you have a larger specific volume, it means each kilogram of the substance takes up more space. This is particularly important in steam and gas calculations. The typical units for specific volume are ${\text{m}}^3/\text{kg}$. If you're using water property tables, you often need to check whether the conditions fall under compressed liquid, saturated, or superheated vapor states. To determine the specific volume for given states like in the exercise, navigate through the steam tables or use software tools designed for property calculations.

Understanding Internal Energy

Internal energy ($u$) represents the total energy related to the molecular structure of a substance, including kinetic and potential energies at the molecular level. It is a key concept in the first law of thermodynamics, where changes in internal energy are connected to heat transfer and work done. For instance, in the exercise, internal energy was to be found for various states like at pressures of 3 bar and temperatures of 240°C. To find this, specific internal energy data can be derived from the steam tables for water. Internal energy values are typically expressed in $\text{kJ}/\text{kg}$. Knowing internal energy helps in understanding heat interactions and work potential of a system. Keep in mind that internal energy changes based on whether a substance is in the liquid phase, vapor phase, or mixture.

Enthalpy in Thermodynamics

Enthalpy ($h$) is another vital thermodynamic property that encompasses the total heat content of a system. It is defined as the sum of internal energy and the product of pressure and volume: $$h=u+p\ast v$$Enthalpy is particularly useful in processes involving heat transfer under constant pressure, such as phase changes from liquid to gas. In the exercise, finding enthalpy at conditions like 400°C and 10 bar involves looking up superheated steam tables. The values are generally given in $\text{kJ}/\text{kg}$. Understanding enthalpy allows us to analyze energy exchanges during thermodynamic cycles and different states. For a given temperature and pressure, you can use specialized tables to locate the enthalpy directly or calculate it based on internal energy and specific volume.