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 need to determine the heat transfer rate through the plate using Fourier's law of heat conduction:

Q = k * A * (T_h - T_c) / L

Where:
Q is the heat transfer rate
k is the thermal conductivity (4 W/mK)
A is the area of the face of the plate through which heat is being transferred (20 cm * 20 cm = 400 cm^2 = 0.04 m^2)
T_h is the temperature at the hot boundary (325⁰C)
T_c is the temperature at the cold boundary (25⁰C)
L is the distance between the hot and cold boundaries (10 cm = 0.1 m)

First, convert the temperatures to kelvin (K):
T_h = 325 + 273 = 598 K
T_c = 25 + 273 = 298 K

Now, calculate the heat transfer rate:
Q = 4 * 0.04 * (598 - 298) / 0.1 = 4 * 0.04 * 300 / 0.1 = 480 W

Therefore, the total amount of heat injected into the plate from the top edge is 480 watts.