Answer: To solve this problem, we need to use the conservation of energy equation, which states that the rate of heat transfer into a system is equal to the rate of internal energy generation plus the rate of heat transfer out of the system.
Explanation:
a) The rate of heat transfer into the system is equal to the rate at which air flows through the duct, multiplied by the specific heat of air, multiplied by the temperature difference between the inlet and outlet:
Q_in = m_dot * cp * (T_out - T_in)
where Q_in is the rate of heat transfer into the system, m_dot is the mass flow rate of air, cp is the specific heat of air, T_out is the temperature of air at the outlet, and T_in is the temperature of air at the inlet.
The mass flow rate of air is given by:
m_dot = rho * A * v
where rho is the density of air, A is the cross-sectional area of the duct, and v is the velocity of air.
The velocity of air can be calculated from the volumetric flow rate and the cross-sectional area of the duct:
v = Q_in / (A * V_dot)
where V_dot is the volumetric flow rate of air.
Using the given values, we have:
m_dot = rho * A * v = 1.2 kg/m^3 * pi * (0.15 m / 2)^2 * (0.65 m^3/min / 60 s/min) = 0.0129 kg/s
v = Q_in / (A * V_dot) = (0.65 m^3/min / 60 s/min) / (pi * (0.15 m / 2)^2) = 3.75 m/s
The rate of heat transfer into the system is:
Q_in = m_dot * cp * (T_out - T_in) = 0.0129 kg/s * 1005 J/(kg*K) * (T_out - 27°C)
The rate of internal energy generation is equal to 85% of the total heat generated by the components:
Q_gen = 0.85 * 220 W = 187 W
The rate of heat transfer out of the system is equal to the rate of heat loss through the outer surfaces of the duct, which can be calculated using the thermal resistance of the duct:
Q_out = (T_s - T_inf) / R_th
where T_s is the surface temperature of the duct, T_inf is the ambient temperature, and R_th is the thermal resistance of the duct. The thermal resistance of the duct can be calculated using the thermal conductivity and thickness of the duct:
R_th = ln(r2 / r1) / (2pik*L)
where r1 and r2 are the inner and outer radii of the duct, k is the thermal conductivity of the duct material, and L is the length of the duct.
Using the given values, we have:
R_th = ln(0.15/2 / 0.15/2 - 0.003) / (2pi0.15 W/(m*K) * 1 m) = 0.0096 K/W
Q_out = (T_s - T_inf) / R_th = (T_s - 27°C) / 0.0096 K/W
Since the total rate of heat transfer out of the system must equal the total rate of heat transfer into the system, we can set the two equations equal to each other:
m_dot * cp * (T_out - T_in) = Q_gen + Q_out
Substituting in the values we calculated, we have:
0.0129 kg/s * 1005 J/(kg*K)
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