A hollow tube having an inside diameter of 2.5 cm and a wall thickness of 0.4 mm is exposed to an environment at h=100 W/m2¨¬C and T¡Ä =40¨¬C. What heat generation rate in the tube will produce a maximum tube temperature of 250¨¬C for k =24 W/m¨¬C?
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To determine the heat generation rate in the tube, we can use the formula for heat conduction:
Q = k * A * (T2 - T1) / L
Where:
Q is the heat generation rate in watts (W)
k is the thermal conductivity of the material in watts per meter-kelvin (W/m·K)
A is the surface area through which heat is conducted in square meters (m²)
T2 is the maximum tube temperature in degrees Celsius (°C)
T1 is the environment temperature in degrees Celsius (°C)
L is the thickness of the tube in meters (m)
First, let's calculate the surface area through which heat is conducted.
The outside diameter of the tube can be calculated by adding twice the wall thickness to the inside diameter:
Outside diameter = Inside diameter + 2 * Wall thickness
Outside diameter = 2.5 cm + 2 * 0.4 mm
Outside diameter = 2.5 cm + 0.8 mm
Outside diameter = 2.5 cm + 0.08 cm
Outside diameter = 2.58 cm
The surface area (A) can be calculated using the formula for the surface area of a cylinder:
A = 2 * π * (Outside radius * Height)
A = 2 * π * (Outside diameter / 2) * (Height of the tube)
A = π * (Outside diameter) * (Height of the tube)
Since the tube is hollow, the height of the tube is not given. We'll assume it to be 1 meter for simplicity.
A = π * 2.58 cm * 100 cm
A = 258 cm²
A = 0.0258 m²
Now, let's calculate the heat generation rate (Q) using the formula mentioned earlier.
Q = k * A * (T2 - T1) / L
Q = 24 W/m·K * 0.0258 m² * (250°C - 40°C) / 0.004 m
Q = 24 W/m·K * 0.0258 m² * 210°C / 0.004 m
Q = 24 W/m·K * 0.0258 m² * 5235 K/m
Q = 318.28 W
Therefore, a heat generation rate of approximately 318.28 watts will produce a maximum tube temperature of 250°C.
To find the heat generation rate in the tube that will produce a maximum temperature of 250°C, we need to consider the heat transfer mechanisms occurring in the system.
First, we need to calculate the surface area of the tube. The outer diameter of the tube is the sum of the inner diameter and twice the wall thickness:
Outer diameter = Inner diameter + 2 * wall thickness
= 2.5 cm + 2 * 0.4 mm
Converting the inner diameter and wall thickness to meters:
Inner diameter = 2.5 cm / 100
Wall thickness = 0.4 mm / 1000
Outer diameter = (2.5 / 100 + 2 * 0.4 / 1000) meters
Next, we can calculate the surface area of the tube:
Surface area = 2 * π * (Outer radius)^2
where Outer radius = Outer diameter / 2
Now, we can determine the heat flux in the tube using Newton's law of cooling:
Q = h * A * (Tmax - T∞)
where Q is the heat generation rate, h is the heat transfer coefficient, A is the surface area, Tmax is the maximum temperature, and T∞ is the ambient temperature.
Given:
h = 100 W/m2°C
T∞ = 40°C
Tmax = 250°C
Now, we have all the necessary values to calculate the heat generation rate:
Q = 100 * A * (250 - 40)
Remember, we still need to find the surface area (A) to complete the calculation.
Plugging in the values and solving the equation will give us the heat generation rate required to produce a maximum tube temperature of 250°C.