You have a vertical plate which is 20 cm square and 0.75 cm thick. The plate has a thermal conductivity of 4 W/mK. The top edge of the plate is kept at a temperature of 325⁰C, and the bottom edge is kept at a temperature of 275⁰C. The vertical edges of the plate are insulated, but the faces of the plate are in contact with a 25⁰C fluid through a film coefficient of 10 W/m2K. What is the total amount of heat injected into the plate from the top edge?
the nodes are 10 cm apart
To calculate the total amount of heat injected into the plate from the top edge, we can use the one-dimensional heat conduction equation:
Q = k * A * (T_top - T_bottom) / L
Where:
Q is the heat flow
k is the thermal conductivity of the plate
A is the cross-sectional area of the plate
T_top is the temperature at the top edge
T_bottom is the temperature at the bottom edge
L is the thickness of the plate
First, let's calculate the cross-sectional area of the plate:
A = width * height = (20 cm) * (0.75 cm) = 15 cm^2 = 0.0015 m^2
Next, let's convert the temperatures to Kelvin:
T_top = 325⁰C + 273.15 = 598.15 K
T_bottom = 275⁰C + 273.15 = 548.15 K
Substituting the given values into the equation:
Q = (4 W/mK) * (0.0015 m^2) * (598.15 K - 548.15 K) / (0.75 cm)
Since the nodes are 10 cm apart, the thickness of each node will be L = 10 cm / 2 = 5 cm = 0.05 m.
Q = (4 W/mK) * (0.0015 m^2) * (598.15 K - 548.15 K) / (0.05 m)
Q = 0.00006 W * (598.15 K - 548.15 K)
Q = 0.00006 W * 50 K
Q = 0.003 W
Therefore, the total amount of heat injected into the plate from the top edge is 0.003 W.
To find the total amount of heat injected into the plate from the top edge, we can use the concept of one-dimensional heat conduction through a solid plate.
First, we need to calculate the temperature distribution along the plate. Since the plate is one-dimensional, it can be divided into several "nodes" or sections, with each node representing a specific distance along the plate. Given that the distance between each node is 10 cm (0.1 m), we can calculate the number of nodes in the plate.
The total length of the plate is 0.75 cm, so the number of nodes can be calculated as:
Number of nodes = Total length / Distance between nodes
Number of nodes = 0.75 cm / 10 cm
Number of nodes = 0.075
Since the number of nodes is not a whole number, we can assume there are 8 nodes distributed along the plate.
Next, we need to calculate the temperature at each of these nodes using the concept of steady-state heat conduction.
Starting with the top edge temperature of 325⁰C and the bottom edge temperature of 275⁰C, we can assume that the temperature at each node is constant. Therefore, the temperature difference between each node can be determined by dividing the temperature gradient by the number of nodes.
Temperature difference between each node = (Top edge temperature - Bottom edge temperature) / Number of nodes
Temperature difference between each node = (325⁰C - 275⁰C) / 8
Temperature difference between each node = 50⁰C / 8
Temperature difference between each node = 6.25⁰C
Now we can calculate the temperature at each node by starting from the top edge temperature and subtracting the temperature difference.
Temperature at Node 1 = Top edge temperature = 325⁰C
Temperature at Node 2 = Top edge temperature - Temperature difference between each node = 325⁰C - 6.25⁰C = 318.75⁰C
Temperature at Node 3 = 318.75⁰C - 6.25⁰C = 312.5⁰C
Temperature at Node 4 = 312.5⁰C - 6.25⁰C = 306.25⁰C
Temperature at Node 5 = 306.25⁰C - 6.25⁰C = 300⁰C
Temperature at Node 6 = 300⁰C - 6.25⁰C = 293.75⁰C
Temperature at Node 7 = 293.75⁰C - 6.25⁰C = 287.5⁰C
Temperature at Node 8 (Bottom edge temperature) = 275⁰C
Now that we have the temperature distribution along the plate, we can calculate the heat transferred through each node using Fourier's law of heat conduction.
Heat transferred through each node = (Thermal conductivity x Area x Temperature difference) / Distance between nodes
The area of each node is the same, as the plate is of constant cross-sectional area (20 cm x 20 cm = 400 cm² = 0.04 m x 0.04 m = 0.0016 m²).
Heat transferred through each node = (4 W/mK x 0.0016 m² x Temperature difference) / 0.1 m
Heat transferred through each node = 0.064 x Temperature difference
Now we can calculate the total amount of heat injected into the plate from the top edge by summing up the heat transferred through each node.
Total heat injected = Heat transferred through Node 1 + Heat transferred through Node 2 + ... + Heat transferred through Node 7 + Heat transferred through Node 8
Total heat injected = (0.064 x Temperature difference) + (0.064 x Temperature difference) + ... + (0.064 x Temperature difference)
Total heat injected = 8 x (0.064 x Temperature difference)
Total heat injected = 8 x (0.064 x 6.25⁰C)
Total heat injected = 3.2 W
Therefore, the total amount of heat injected into the plate from the top edge is 3.2 Watts.