A stream of air flows in a duct of 100 mm diameter at a rate of 1 kg/s.the stagnation
temperature is 37 °C.at one section of the duct the static pressure is 40 kPa.Calculate the Mach
number ,velocity,and the stagnation pressure at this section.
Answers
Answer:
500-200 =300
Explanation:
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Answer:
P0 = 40,000 (initial guess)
P0_new = P / (1 + 1.4/2 * 0.4^2)^(1.4
Explanation:
To solve this problem, we can use the conservation equations for mass, momentum, and energy, along with the ideal gas law. The Mach number is given by the ratio of the flow velocity to the local speed of sound, which can be calculated using the static temperature and pressure.
Given:
Duct diameter = 100 mm
Air mass flow rate = 1 kg/s
Stagnation temperature (T0) = 37 °C = 310.15 K
Static pressure (P) = 40 kPa = 40,000 Pa
We can assume that the air is an ideal gas, which means that the following equation applies:
P = ρRT
where P is the static pressure, ρ is the density, R is the gas constant, and T is the static temperature. Rearranging this equation, we get:
ρ = P / (RT)
Using the ideal gas law, we can express the density in terms of the stagnation temperature and pressure:
ρ0 = P0 / (RT0)
where P0 is the stagnation pressure. Assuming adiabatic and isentropic flow, the stagnation pressure can be related to the static pressure and the Mach number using the following equation:
P0 / P = (1 + γ/2 * (γ-1)/2 * M^2)^(γ/(γ-1))
where γ is the ratio of specific heats of air at constant pressure and constant volume (γ = 1.4 for air). This equation can be solved iteratively to obtain the Mach number corresponding to the given static pressure.
First, we can calculate the density of the air using the given static pressure and temperature:
ρ = P / (RT) = 40,000 / (287.058 * 310.15) = 1.384 kg/m^3
The mass flow rate can be expressed in terms of the density and velocity:
m_dot = ρ * A * V
where A is the cross-sectional area of the duct (πD^2/4):
A = π/4 * (0.1 m)^2 = 0.007853 m^2
V = m_dot / (ρ * A) = 1 / (1.384 * 0.007853) = 128.25 m/s
The speed of sound can be calculated using the following equation:
a = sqrt(γ * R * T)
where R is the gas constant for air (287.058 J/kg K). Evaluating this equation at the given static temperature, we get:
a = sqrt(1.4 * 287.058 * 310.15) = 348.97 m/s
Therefore, the Mach number is:
M = V / a = 128.25 / 348.97 = 0.3678
Finally, we can solve for the stagnation pressure using the equation given above. Since this equation cannot be solved analytically, we can use an iterative method such as the Newton-Raphson method to obtain a numerical solution. Here, we will use a simpler fixed-point iteration method, which involves rearranging the equation as follows:
P0 = P / (1 + γ/2 * (γ-1)/2 * M^2)^(γ/(γ-1))
Starting with an initial guess of P0 = P, we can iterate using this equation until the difference between successive values is less than some tolerance (e.g. 0.01%). Here, we will perform 10 iterations:
P0 = 40,000 (initial guess)
P0_new = P / (1 + 1.4/2 * 0.4^2)^(1.4
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