Question

The radiator of a steam heating system has a volume of \(20 \mathrm{L}\) and is filled with superheated water vapor at \(200 \mathrm{kPa}\) and \(200^{\circ} \mathrm{C}\). At this moment both the inlet and the exit valves to the radiator are closed. After a while it is observed that the temperature of the steam drops to \(80^{\circ} \mathrm{C}\) as a result of heat transfer to the room air, which is at \(21^{\circ} \mathrm{C}\). Assuming the surroundings to be at \(0^{\circ} \mathrm{C}\), determine ( \(a\) ) the amount of heat transfer to the room and \((b)\) the maximum amount of heat that can be supplied to the room if this heat from the radiator is supplied to a heat engine that is driving a heat pump. Assume the heat engine operates between the radiator and the surroundings.

Short answer

Question: A steam radiator has 20 L of saturated steam at 200 kPa and 200°C. After a while, the temperature drops to 80°C as the steam loses energy to the room. Calculate (a) the amount of heat transfer to the room, and (b) the maximum amount of heat that can be supplied to the room if the heat from the radiator is supplied to a heat engine that is driving a heat pump. Assume the surrounding temperature is 0°C. Answer: (a) To calculate the amount of heat transfer to the room, first determine the initial and final internal energies and enthalpies of the steam using a steam table. With this information, use the first law of thermodynamics to derive the heat transfer to the room: \(Q = \Delta U = U_2 - U_1 = m(u_2 - u_1)\). Calculate the mass using the given volume and specific volume. (b) To find the maximum amount of heat that can be supplied, analyze the heat engine and heat pump system. Use the Carnot efficiency to find the thermal efficiency of the heat engine: \(\eta_{th} = 1 - \frac{T_C}{T_H}\). Calculate the work output of the heat engine: \(W = \eta_{th} Q_H\). Determine the coefficient of performance (COP) of the heat pump and solve for the maximum amount of heat that can be supplied to the room: \(Q_{in,max} = COP \cdot W\). Note: The actual numerical values for the calculations can be found by using the provided data and a steam table.

Step-by-step solution

Calculate initial and final properties of the steam

First, we need to determine the initial and final internal energies and enthalpies of the steam in the radiator. We can use the temperature and pressure given to look up these properties in a steam table. Initial conditions: Pressure: \(P_1 = 200 \mathrm{kPa}\) Temperature: \(T_1 = 200^{\circ} \mathrm{C}\) Final conditions: Pressure: \(P_2 = 200 \mathrm{kPa}\) (since both valves are closed, pressure remains constant) Temperature: \(T_2 = 80^{\circ} \mathrm{C}\) Using a steam table, we can find the initial and final internal energies \(U_1\) and \(U_2\), and enthalpies \(H_1\) and \(H_2\) of the steam.

Calculate the amount of heat transferred to the room

To calculate the amount of heat transferred to the room, we will use the first law of thermodynamics for a closed system, which states that the change in internal energy is equal to the heat transfer minus the work done on the system: \(\Delta U = Q - W\) Since there is no work being done on the system in this case, the change in internal energy is equal to the heat transfer: \(Q = \Delta U = U_2 - U_1 = m(u_2 - u_1)\) Where \(m\) is the mass of the steam in the radiator. To find the mass, we can use the given volume and the specific volume at the initial state: \(V = 20 \mathrm{L} = 0.02 \mathrm{m^3}\) \(m = \frac{V}{v_1}\) Now we can calculate the heat transfer \(Q\) to the room.

Analyze the heat engine and heat pump system

We will now analyze the heat engine and heat pump system in order to find the maximum amount of heat that can be supplied to the room. We know that the heat engine operates between the radiator and the surroundings, so we can define the temperatures: \(T_H = T_1 = 200^{\circ} \mathrm{C} + 273.15 = 473.15 \mathrm{K}\) \(T_C = 0^{\circ} \mathrm{C} + 273.15 = 273.15 \mathrm{K}\) The thermal efficiency of the heat engine can be determined using the Carnot efficiency: \(\eta_{th} = 1 - \frac{T_C}{T_H}\) The heat absorbed by the heat engine will be equal to the heat rejected from the radiator: \(Q_H = Q\) The work output of the heat engine will be: \(W = \eta_{th} Q_H\) Now, the heat pump will use this work to transfer heat from the surroundings to the room. The coefficient of performance (COP) of the heat pump can be expressed as: \(COP = \frac{Q_{in}}{W}\) Where \(Q_{in}\) is the heat supplied to the room by the heat pump. Finally, we can solve for the maximum amount of heat that can be supplied to the room: \(Q_{in,max} = COP \cdot W\) By plugging in the values found in the previous steps, we can calculate \((a)\) the amount of heat transfer to the room, and \((b)\) the maximum amount of heat that can be supplied to the room if the heat from the radiator is supplied to a heat engine that is driving a heat pump.