Air flows along a 1 m long plate. If the temperature of flow is 0 deg Celsius and the surface temperature of plate is maintained at 30 deg Celsius. Evaluate the heat flow from the plate to air. What are the maximum thickness of the velocity boundary layer and thermal boundary layer? Where do they appear?
To evaluate the heat flow from the plate to the air, we need to calculate the convective heat transfer coefficient (h) and then use it in the heat transfer equation.
To calculate h, we can use a correlation called the Nusselt number (Nu). The Nusselt number will depend on the flow conditions and the geometry of the plate. In this case, since the flow is along the plate, we can use the correlation for flow over a flat plate.
The Nusselt number for flow over a flat plate can be calculated using the Reynolds number (Re) and the Prandtl number (Pr). The Reynolds number is a measure of the flow regime, and the Prandtl number is a measure of the fluid's thermal properties.
To determine the Reynolds number, we need to know the velocity of the air flowing along the plate. Let's assume a velocity of 10 m/s.
The Prandtl number for air can be considered constant at 0.7.
The Reynolds number (Re) is calculated using the equation:
Re = (velocity * characteristic length) / kinematic viscosity
The characteristic length for flow along a flat plate is usually taken as the distance from the leading edge of the plate to the point of interest. In this case, it is 1 meter.
The kinematic viscosity (v) for air at 0°C can be found in literature and is approximately 1.57 x 10^-5 m^2/s.
Now we can calculate the Reynolds number:
Re = (10 * 1) / (1.57 x 10^-5) = 6.37 x 10^5
Next, we can use the Reynolds number and Prandtl number to find the Nusselt number through empirical correlations. Let's use the correlation for flow over a flat plate:
Nu = 0.332 * (Re^0.5) * (Pr^(1/3))
Plugging in the values, we get:
Nu = 0.332 * (6.37 x 10^5)^0.5 * (0.7^(1/3)) ≈ 322.5
Finally, to calculate the convective heat transfer coefficient (h), we use the equation:
h = (Nu * thermal conductivity) / characteristic length
For air at 0°C, the thermal conductivity can be found in literature and is approximately 0.0257 W/(m*K).
h = (322.5 * 0.0257) / 1 = 8.3 W/(m^2*K)
Now that we know the convective heat transfer coefficient (h), we can use it in the heat transfer equation to evaluate the heat flow from the plate to the air. The equation is:
Q = h * A * (Ts - Tf)
Where:
Q is the heat flow (W)
h is the convective heat transfer coefficient (W/(m^2*K))
A is the surface area of the plate (m^2)
Ts is the surface temperature of the plate (30°C)
Tf is the temperature of the flow (0°C)
To calculate the surface area (A), we use the length of the plate (1 m) and its width (which is not given). Let's assume a width of 0.1 m.
A = length * width = 1 * 0.1 = 0.1 m^2
Q = 8.3 * 0.1 * (30 - 0) = 24.9 W
Therefore, the heat flow from the plate to the air is 24.9 Watts.
Now, let's calculate the maximum thickness of the velocity boundary layer (δ_v) and the thermal boundary layer (δ_t), and determine where they appear.
The maximum thickness of the velocity boundary layer can be approximated using the formula:
δ_v ≈ 5 * (x / Re_x) ^ (1/2)
Where:
x is the distance from the leading edge of the plate (in this case, 1 m)
Re_x is the Reynolds number at x.
The Reynolds number at x can be calculated as follows:
Re_x = (velocity * x) / kinematic viscosity
Re_x = (10 * 1) / (1.57 x 10^-5) = 6.37 x 10^5
Substituting these values into the formula, we get:
δ_v ≈ 5 * (1 / (6.37 x 10^5))^(1/2) ≈ 4.98 x 10^-3 m ≈ 5 mm
Therefore, the maximum thickness of the velocity boundary layer is approximately 5 mm and it appears along the length of the plate.
Similarly, the maximum thickness of the thermal boundary layer can be approximated using the formula:
δ_t ≈ 5 * (x / Re_x) ^ (1/2)
Using the same values of x and Re_x as before, we get:
δ_t ≈ 5 * (1 / (6.37 x 10^5))^(1/2) ≈ 4.98 x 10^-3 m ≈ 5 mm
Hence, the maximum thickness of the thermal boundary layer is approximately 5 mm and it also appears along the length of the plate.