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An air-conditioning system involves the mixing of cold air and warm outdoor before the mixture is routed to the conditional room in steady operation. Cold air enters the mixing chamber at 7 C and 105kpa at a rate of 0. 55 m3/s while warm air enters at 34 C and 105 kpa. The air leaves the room at 24 C.

The ratio of the mass flow rates of the hot to cold air steams is 1. 6

using variable specific heats, determine

a) the mixture temperture at the inlet of the room

b) the rate of heat gain of the room

User Pyjavo
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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.
User ChrisProsser
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