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?
To solve this problem, we need to use the heat transfer equation:
Q = U x A x ∆T
Where:
Q = Heat generated
U = Overall heat transfer coefficient
A = Surface area of the tube
∆T = Temperature difference
To find the value of U, we need to consider both conduction and convection heat transfer. The formula for the overall heat transfer coefficient is:
1/U = 1/hi + (t/k) + 1/ho
Where:
hi = Inside heat transfer coefficient
t = Thickness of the tube wall
k = Thermal conductivity of the tube material
ho = Outside heat transfer coefficient
To simplify the problem, we can assume that the heat transfer coefficients on both sides are equal (hi = ho). Also, neglecting radiation, we can ignore the third term in the equation:
1/U = 1/hi + (t/k)
Now, let's calculate each component step by step.
Inside heat transfer coefficient (hi):
The inside heat transfer coefficient depends on the fluid inside the tube. Since the fluid is not specified, we can assume air flowing over the inside surface of the tube. To find hi, we can use empirical correlations or look up values in tables based on the flow conditions and tube geometry.
For this case, let's assume a forced convection flow with air inside the tube at an average velocity. Using correlations, hi can be estimated to be around 50 W/m²·°C.
Outside heat transfer coefficient (ho):
The outside heat transfer coefficient depends on the heat transfer conditions on the outside of the tube. Since it is exposed to an environment, we can assume natural convection. Again, depending on the conditions (e.g., vertical or horizontal orientation), the value can be estimated.
For this case, we will assume a vertical orientation. Based on empirical correlations, we can estimate ho to be around 15 W/m²·°C.
Now we can calculate 1/U:
1/U = 1/50 + (0.004/24) + 1/15
From this, we can find the value of U: U = 1 / (1/U)
Next, we need to calculate the surface area of the tube.
Surface area (A):
The surface area of the tube can be found using the formula:
A = π x (D - t) x L
Where:
D = Inside diameter
t = Wall thickness
L = Length of the tube
For this problem, the values given are:
Inside diameter (D) = 2.5 cm = 0.025 m
Wall thickness (t) = 0.4 mm = 0.0004 m
Let's assume a length (L) of 1 meter.
Now we can calculate the surface area:
A = π x (0.025 - 0.0004) x 1
Finally, we can calculate the heat generation rate (Q) using the heat transfer equation:
Q = U x A x ∆T
For this problem, the maximum tube temperature is given as 250°C, and the ambient temperature is 40°C, so the temperature difference (∆T) is 250 - 40 = 210°C.
Plug in the values of U, A, and ∆T into the equation to find Q:
Q = U x A x ∆T
By substituting the values and solving the equation, you can find the heat generation rate (Q) required to produce a maximum tube temperature of 250°C.