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Sagot :
Sure, let's work through both parts of this problem step-by-step.
Here are the known values:
- Temperature of hot gases (\( T_{\text{gas}} \)) = \( 280^{\circ}C \)
- Temperature of air (\( T_{\text{air}} \)) = \( 35^{\circ}C \)
- Heat transfer coefficient of hot gases (\( h_{\text{gas}} \)) = \( 31.5 \text{ W/m}^2\text{K} \)
- Heat transfer coefficient of air (\( h_{\text{air}} \)) = \( 32 \text{ W/m}^2\text{K} \)
- Thermal conductivity of the metal plate (\( k_{\text{plate}} \)) = \( 50 \text{ W/mK} \)
- Thickness of the metal plate (\( t_{\text{plate}} \)) = \( 0.01 \text{ m} \)
### Part (i): Calculate the overall heat transfer coefficient (U)
The overall heat transfer coefficient \( U \) for a composite system like this can be calculated using the formula:
[tex]\[ \frac{1}{U} = \frac{1}{h_{\text{gas}}} + \frac{t_{\text{plate}}}{k_{\text{plate}}} + \frac{1}{h_{\text{air}}} \][/tex]
Now substituting in the given values:
[tex]\[ \frac{1}{U} = \frac{1}{31.5} + \frac{0.01}{50} + \frac{1}{32} \][/tex]
Calculate each term individually:
[tex]\[ \frac{1}{31.5} \approx 0.03175 \][/tex]
[tex]\[ \frac{0.01}{50} = 0.0002 \][/tex]
[tex]\[ \frac{1}{32} \approx 0.03125 \][/tex]
So,
[tex]\[ \frac{1}{U} = 0.03175 + 0.0002 + 0.03125 = 0.0632 \][/tex]
Now, take the reciprocal to find \( U \):
[tex]\[ U = \frac{1}{0.0632} \approx 15.82378 \text{ W/m}^2\text{K} \][/tex]
### Part (ii): Calculate the heat transfer (Q)
The heat transfer \( Q \) is given by the formula:
[tex]\[ Q = U \times A \times \Delta T \times t \][/tex]
Where:
- \( U \) = \( 15.82378 \text{ W/m}^2\text{K} \)
- Assuming the area \( A \) of the plate to be \( 1 \text{ m}^2 \)
- \( \Delta T \) is the temperature difference between the hot gases and the air:
[tex]\[ \Delta T = T_{\text{gas}} - T_{\text{air}} = 280 - 35 = 245^{\circ}C \][/tex]
- Time \( t \) = \( 60 \) seconds (since we want the heat transfer per minute)
Substitute these values into the equation:
[tex]\[ Q = 15.82378 \times 1 \times 245 \times 60 \][/tex]
[tex]\[ Q = 232609.54199 \text{ J} \][/tex]
Thus, the calculated values are:
1. The overall heat transfer coefficient \( U \) is approximately \( 15.82378 \text{ W/m}^2\text{K} \).
2. The heat transfer from gases to air per minute per square meter of plate area [tex]\( Q \)[/tex] is approximately [tex]\( 232609.54 \text{ J} \)[/tex].
Here are the known values:
- Temperature of hot gases (\( T_{\text{gas}} \)) = \( 280^{\circ}C \)
- Temperature of air (\( T_{\text{air}} \)) = \( 35^{\circ}C \)
- Heat transfer coefficient of hot gases (\( h_{\text{gas}} \)) = \( 31.5 \text{ W/m}^2\text{K} \)
- Heat transfer coefficient of air (\( h_{\text{air}} \)) = \( 32 \text{ W/m}^2\text{K} \)
- Thermal conductivity of the metal plate (\( k_{\text{plate}} \)) = \( 50 \text{ W/mK} \)
- Thickness of the metal plate (\( t_{\text{plate}} \)) = \( 0.01 \text{ m} \)
### Part (i): Calculate the overall heat transfer coefficient (U)
The overall heat transfer coefficient \( U \) for a composite system like this can be calculated using the formula:
[tex]\[ \frac{1}{U} = \frac{1}{h_{\text{gas}}} + \frac{t_{\text{plate}}}{k_{\text{plate}}} + \frac{1}{h_{\text{air}}} \][/tex]
Now substituting in the given values:
[tex]\[ \frac{1}{U} = \frac{1}{31.5} + \frac{0.01}{50} + \frac{1}{32} \][/tex]
Calculate each term individually:
[tex]\[ \frac{1}{31.5} \approx 0.03175 \][/tex]
[tex]\[ \frac{0.01}{50} = 0.0002 \][/tex]
[tex]\[ \frac{1}{32} \approx 0.03125 \][/tex]
So,
[tex]\[ \frac{1}{U} = 0.03175 + 0.0002 + 0.03125 = 0.0632 \][/tex]
Now, take the reciprocal to find \( U \):
[tex]\[ U = \frac{1}{0.0632} \approx 15.82378 \text{ W/m}^2\text{K} \][/tex]
### Part (ii): Calculate the heat transfer (Q)
The heat transfer \( Q \) is given by the formula:
[tex]\[ Q = U \times A \times \Delta T \times t \][/tex]
Where:
- \( U \) = \( 15.82378 \text{ W/m}^2\text{K} \)
- Assuming the area \( A \) of the plate to be \( 1 \text{ m}^2 \)
- \( \Delta T \) is the temperature difference between the hot gases and the air:
[tex]\[ \Delta T = T_{\text{gas}} - T_{\text{air}} = 280 - 35 = 245^{\circ}C \][/tex]
- Time \( t \) = \( 60 \) seconds (since we want the heat transfer per minute)
Substitute these values into the equation:
[tex]\[ Q = 15.82378 \times 1 \times 245 \times 60 \][/tex]
[tex]\[ Q = 232609.54199 \text{ J} \][/tex]
Thus, the calculated values are:
1. The overall heat transfer coefficient \( U \) is approximately \( 15.82378 \text{ W/m}^2\text{K} \).
2. The heat transfer from gases to air per minute per square meter of plate area [tex]\( Q \)[/tex] is approximately [tex]\( 232609.54 \text{ J} \)[/tex].
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