Step-by-step explanation:
To calculate the heat transfer per unit length of the tube, we can use the heat transfer rate equation:
Q = h * A * (T_wall - T_air)
where:
Q is the heat transfer rate per unit length (W/m)
h is the convective heat transfer coefficient (W/m²·°C)
A is the surface area of the tube (m²)
T_wall is the wall temperature (°C)
T_air is the air temperature (°C)
First, let's calculate the convective heat transfer coefficient (h) using the Dittus-Boelter correlation for turbulent flow in a tube:
Nu = 0.023 * Re^0.8 * Pr^0.3
where:
Nu is the Nusselt number
Re is the Reynolds number
Pr is the Prandtl number
The Reynolds number can be calculated as:
Re = (rho * V * D) / mu
where:
rho is the density of the air (kg/m³)
V is the velocity of the air (m/s)
D is the diameter of the tube (m)
mu is the dynamic viscosity of the air (Pa·s)
The Prandtl number for air can be approximated as Pr ≈ 0.7.
Given:
Pressure (P) = 2 atm = 2 * 101325 Pa (converted to SI units)
Temperature (T_air) = 200 °C = 473 K
Velocity (V) = 20 m/s
Diameter (D) = 5 cm = 0.05 m
Wall temperature (T_wall) = T_air + 20 °C = 493 K
First, we need to calculate the air properties at the given conditions:
Using the ideal gas law: PV = nRT
where R is the gas constant for air.
n = (P * V) / (R * T_air)
The dynamic viscosity (mu) can be calculated using Sutherland's formula:
mu = mu_ref * (T_air / T_ref)^(3/2) * (T_ref + S) / (T_air + S)
where:
mu_ref is the reference viscosity (taken at 273.15 K)
T_ref is the reference temperature (taken as 273.15 K)
S is the Sutherland's constant for air (taken as 110 K)
Now, we can calculate the Reynolds number (Re), Prandtl number (Pr), and Nusselt number (Nu) using the given equations.
Finally, we can calculate the heat transfer rate per unit length (Q) using the equation mentioned above.
Please note that obtaining precise values for properties like dynamic viscosity and heat transfer coefficients may require more accurate correlations and experimental data.