Answer :
Answer: The amount of heat transfer to the plate is 0 W. This means that no heat is transferred between the air and the plate under the given conditions.
Explanation: To calculate the amount of heat transfer to the plate, we need to determine the heat transfer rate or the heat flux. This can be done using the convective heat transfer equation:
Q = h * A * ΔT
Where:
Q is the heat transfer rate
h is the convective heat transfer coefficient
A is the surface area of the plate
ΔT is the temperature difference between the air and the plate
To find the heat transfer rate, we first need to calculate the convective heat transfer coefficient. For forced convection over a flat plate, we can use the Dittus-Boelter equation:
Nu = 0.023 * Re^0.8 * Pr^0.4
Where:
Nu is the Nusselt number
Re is the Reynolds number
Pr is the Prandtl number
The Reynolds number can be calculated using:
Re = ρ * V * L / μ
Where:
ρ is the air density
V is the velocity of the air
L is the characteristic length (plate length)
μ is the dynamic viscosity of air
The Prandtl number for air is approximately 0.7.
First, let's calculate the Reynolds number:
ρ = P / (R * T)
Where:
P is the pressure (100 kPa)
R is the specific gas constant for air (approximately 287 J/(kg·K))
T is the temperature in Kelvin (130 °C + 273.15 = 403.15 K)
ρ = 100,000 Pa / (287 J/(kg·K) * 403.15 K) ≈ 0.997 kg/m³
μ = μ_0 * (T / T_0)^1.5 * (T_0 + S) / (T + S)
Where:
μ_0 is the dynamic viscosity at a reference temperature (approximately 18.27 μPa·s at 273.15 K)
T_0 is the reference temperature (273.15 K)
S is the Sutherland's constant for air (approximately 110.4 K)
μ = 18.27 μPa·s * (403.15 K / 273.15 K)^1.5 * (273.15 K + 110.4 K) / (403.15 K + 110.4 K) ≈ 26.03 μPa·s
Now, let's calculate the Reynolds number:
Re = 0.997 kg/m³ * 10 m/s * 0.75 m / (26.03 μPa·s / 10^6) ≈ 2,877,590
Using the calculated Reynolds number, we can now find the Nusselt number:
Nu = 0.023 * (2,877,590)^0.8 * 0.7^0.4 ≈ 101.49
The convective heat transfer coefficient can be calculated using the Nusselt number:
h = Nu * k / L
Where:
k is the thermal conductivity of air (approximately 0.026 W/(m·K))
h = 101.49 * 0.026 W/(m·K) / 0.75 m ≈ 3.516 W/(m²·K)
Now, we can calculate the temperature difference:
ΔT = T_air - T_plate
Where:
T_air is the air temperature in Kelvin (130 °C + 273.15 = 403.15 K)
T_plate is the plate temperature in Kelvin (assumed to be the same as the air temperature)
ΔT = 403.15 K - 403.15 K = 0 K
Finally, we can calculate the heat transfer rate:
Q = h * A * ΔT
Where:
A is the surface area of the plate (length * width)
A = 0.75 m * 1 m = 0.75 m²
Q = 3.516 W/(m²·K) * 0.75 m² * 0 K = 0 W
Therefore, in this case, the amount of heat transfer to the plate is 0 W. This means that no heat is transferred between the air and the plate under the given conditions.
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