Thermodynamics I (Heat Exchanger)

Air from inside a house enters a steady-flow heat exchanger with a volume flow rate of 200 m3/min at 15°C, 100kPa and eaves the heat exchanger at 35°C. This heat exchanger is located inside the air handler of a house heat pump system and receives heat from the heat pump. The heat pump receives heat from air outside the house that acts as a heat source at 10°C.
Assume air has constant specific heats evaluated at 300K.

(a) Determine the heat supplied to the air in the heat exchanger Q. H, in kW.
(b) Assuming the air in the heat exchanger acts as a heat sink having an average temperature of 25°C, determine the minimum power supplied to the heat pump, in kW.

Assumptions:
1. the house air pressure doesn't significantly change due to this process
2. the air is heated at constant pressure as it flows through the heat exchanger
3. the heat exchanger outer walls are well-insulated
4. air's specific heat capacity is constant.

A: To calculate heat supplied to a flowing stream of a fluid with assumed constant specific heats, you use:
Q_dot = m_dot*c*(T2 - T1)

In this example, we are interested in the constant pressure cp of air, because it is free to expand as it flows through the heat exchanger. Pressure is assumed to not change, because we neglect drag losses and there is no other reason for it to change.

Q_dot = m_dot*cp*(T2 - T1)

To get mass flow rate, from volumetric flow rate:
m_dot = V_dot_1*rho_1

To find initial density of air, use this form of the ideal gas law:
rho_1 = P1*M/(R*T1)

M is molar mass of air
P1 is pressure at state 1
T1 is temperature at state 1
R is the universal Avogadro gas constant

Substitute:
Q_dot = V_dot_1*(P1*M/(R*T1)) * cp * (T2 - T1)

Value of cp for air:
cp = k*(R/M)/(k-1)

where k, the adiabatic index, for AIR, is 1.4

Substitute:
Q_dot = V_dot_1*(P1*M/(R*T1)) * ( k*(R/M)/(k-1))* (T2 - T1)