To solve this problem, we can use the principle of energy conservation and the equations for the specific heats of air. Let's go step by step:
a) To find the mixture temperature at the inlet of the room, we can use the equation:
(m_h * T_h + m_c * T_c) / (m_h + m_c) = T_m
where:
m_h = mass flow rate of hot air
T_h = temperature of hot air
m_c = mass flow rate of cold air
T_c = temperature of cold air
T_m = mixture temperature
Given that the ratio of the mass flow rates is 1.6, we can say m_h = 1.6 * m_c. Let's substitute the known values:
(1.6 * m_c * 34 + m_c * 7) / (1.6 * m_c + m_c) = T_m
Simplifying the equation:
(54.4 * m_c + 7 * m_c) / 2.6 * m_c = T_m
(61.4 * m_c) / (2.6 * m_c) = T_m
61.4 / 2.6 = T_m
T_m = 23.62°C
Therefore, the mixture temperature at the inlet of the room is approximately 23.62°C.
b) To calculate the rate of heat gain of the room, we can use the equation:
Q = m_c * c_c * (T_m - T_r)
where:
Q = rate of heat gain
m_c = mass flow rate of cold air
c_c = specific heat of cold air
T_m = mixture temperature
T_r = temperature of the room (leaving air temperature)
The specific heat of air can vary with temperature, but for simplicity, let's assume c_c is constant at room conditions.
Substituting the known values:
Q = 0.55 * c_c * (23.62 - 24)
Simplifying the equation:
Q = -0.55 * c_c
Therefore, the rate of heat gain of the room is -0.55 * c_c. Note that the negative sign indicates a heat loss from the room rather than a gain.
Please note that the specific heat values and units are not provided, so the result for the rate of heat gain is expressed relative to c_c. You would need to know the specific heat value and units to obtain an absolute value.