Get the information you need from a community of experts on IDNLearn.com. Our platform provides detailed and accurate responses from experts, helping you navigate any topic with confidence.
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].
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
