Answer: To solve the problem, we'll go step by step as follows:
(a) Find the velocity of the air through the duct and the Reynolds number:
Velocity (u):
Given: Air enters at 85°C and exits at 70°C, so the bulk mean temperature (T_bulk) = 80°C = 353.15 K.
Given: Duct length (L) = 10 m and volumetric flow rate (Q) = 0.10 m/s.
The volumetric flow rate (Q) is given by Q = u * A, where A is the cross-sectional area of the duct.
Cross-sectional area (A) = 0.15 m * 0.15 m = 0.0225 m²
0.10 m/s = u * 0.0225 m²
u ≈ 4.44 m/s
The velocity of the air through the duct is approximately 4.44 m/s.
Reynolds number (Re):
The Reynolds number is given by Re = (ρ * u * L) / μ
Where:
ρ = density of air at bulk mean temperature (80°C) = density of air at 353.15 K (use tables)
μ = dynamic viscosity of air at bulk mean temperature (80°C) = dynamic viscosity of air at 353.15 K (use tables)
Let's calculate the Reynolds number:
First, we need to find the density (ρ) and dynamic viscosity (μ) of air at 353.15 K (80°C):
From air property tables at 353.15 K:
Density (ρ) ≈ 1.161 kg/m³
Dynamic Viscosity (μ) ≈ 0.0233 Ns/m²
Re = (1.161 kg/m³ * 4.44 m/s * 10 m) / 0.0233 Ns/m² ≈ 23285.415
The Reynolds number (Re) is approximately 23285.415.
(b) Determine the entry length of the flow and the percentage of the entry length from the total duct length:
The entry length (L_e) is the distance over which the flow transitions from fully developed flow to the entrance condition.
For turbulent flow in a square duct, the entry length is given by L_e ≈ 0.06 * Re * H, where H is the hydraulic diameter.
Hydraulic diameter (H) = 4 * (Cross-sectional area / Perimeter) = 4 * (0.0225 m² / 0.6 m) ≈ 0.15 m
L_e ≈ 0.06 * 23285.415 * 0.15 m ≈ 209.56 m
The entry length of the flow is approximately 209.56 meters.
Percentage of entry length from the total duct length:
Percentage = (L_e / L) * 100
Percentage ≈ (209.56 m / 10 m) * 100 ≈ 2095.6%
The entry length represents around 2095.6% of the total duct length.
(c) Use a proper relationship to evaluate the Nusselt number (Nu) and the heat transfer coefficient (h):
For fully developed turbulent flow in a square duct, the Nusselt number is correlated with the Reynolds number by the following empirical relationship:
Nu ≈ 0.023 * Re^0.8 * Pr^0.4
where Pr is the Prandtl number, which can be estimated at the bulk mean temperature (80°C).
From air property tables at 353.15 K:
Pr ≈ 0.715
Nu ≈ 0.023 * (23285.415)^0.8 * (0.715)^0.4 ≈ 83.2
Now, the heat transfer coefficient (h) can be calculated using the relationship:
h = Nu * k / H
where k is the thermal conductivity of air at the bulk mean temperature (80°C).
From air property tables at 353.15 K:
k ≈ 0.0298 W/(m·K)
h ≈ 83.2 * 0.0298 W/(m·K) / 0.15 m ≈ 16.4 W/(m²·K)
The Nusselt number (Nu) is approximately 83.2, and the heat transfer coefficient (h) is approximately 16.4 W/(m²·K).
(d) Determine the exit temperature of the air and the rate of heat loss from the duct to the air space in the attic:
To find the exit temperature of the air, we can use the energy equation:
Q = m * C_p * (T_exit - T_bulk)
where Q is the rate of heat loss from the duct, m is the mass flow rate of air, and C_p is the specific heat capacity of air at constant pressure.
From air property tables at 353.15 K:
C_p ≈ 1005 J/(kg·K)
The mass flow rate (m_dot) can be calculated from the volumetric flow rate (Q) and the density (ρ) at the bulk mean temperature (80°C):
m_dot = Q * ρ ≈ 0.10 m³/s * 1.161 kg/m³ ≈ 0.1161 kg/s
Now, let's find the exit temperature (T_exit):
Q = 0.1161 kg/s * 1005 J/(kg·K) * (T_exit - 353.15 K)
Rate of heat loss from the duct (Q) = 941 W (given)
941 W = 0.1161 kg/s * 1005 J/(kg·K) * (T_exit - 353.15 K)
T_exit - 353.15 K ≈ 941 W / (0.1161 kg/s * 1005 J/(kg·K))
T_exit - 353.15 K ≈ 8.174 K
T_exit ≈ 361.324 K
T_exit ≈ 361.324°C (approximately)
Finally, we can calculate the rate of heat loss from the duct to the air space in the attic:
Rate of heat loss (Q) = m_dot * C_p * (T_exit - T_bulk)
Q ≈ 0.1161 kg/s * 1005 J/(kg·K) * (361.324 K - 353.15 K)
Q ≈ 0.1161 kg/s * 1005 J/(kg·K) * 8.174 K ≈ 941 W (approximately)
The rate of heat loss from the duct to the air space in the attic is approximately 941 watts.