Pure copper wire of diameter = 2.0 mm has an electrical resistance, R, of 0.005 ohm/m (upside down U/m). p = 8900 kg/ m cubed, Cp= 440 J/(kg K). Passage of electrical current produces thermal energy generation = I triangle V= I squared R (J/sec).
Note that Units: Volts = J/coulomb; ohms = volts/amps; amps = coulombs/sec.
If the wire is bare with convective heat flow to the surroundings at T infinity = 25 degrees Celsius with h = 18 W/ (m squared K), calculate the steady-state temperature of the wire.
To calculate the steady-state temperature of the wire, we need to consider both the electrical and thermal properties of the wire.
First, let's calculate the electrical current passing through the wire. The formula relating current (I), voltage (V), and resistance (R) is given as:
I = V / R
From the information given, we know the resistance per unit length of the wire (0.005 Ω/m). However, we need to determine the voltage drop across the wire. This can be done using Ohm's law:
V = I * R
Since the wire is bare and exposed to the surroundings with convective heat flow, the thermal energy generated (I * V) will be dissipated to the surroundings. We can calculate the power generated using the formula:
Power (P) = I^2 * R
Now, let's consider the heat transfer from the wire to the surroundings. The rate of heat transfer, Q, due to convection is given by:
Q = h * A * (T - T_infinity)
where h is the convective heat transfer coefficient, A is the surface area of the wire, T is the temperature of the wire, and T_infinity is the ambient temperature.
Let's assume the wire is of sufficient length so that we can consider it as a long, infinite cylinder. Then the surface area of the wire, A, can be calculated using the formula:
A = 2 * π * r * L
where r is the radius of the wire and L is its length.
Now, let's sum up the power generated by the electrical current and the power dissipated through the convective heat flow:
Power (P) = I^2 * R = Q
Since the steady-state temperature is constant, the heat transfer to the surroundings is equal to the heat generated by the electrical current.
Let's substitute the appropriate equations into the power equation:
I^2 * R = h * A * (T - T_infinity)
Now, we can rearrange the equation to solve for T:
T = (I^2 * R) / (h * A) + T_infinity
Plugging in the values given:
Diameter (d) = 2.0 mm = 2.0 * 10^-3 m
Radius (r) = d /2 = 1.0 * 10^-3 m
Length (L) = Assuming the wire is long enough (considered as infinite)
T_infinity = 25 degrees Celsius = 25 + 273.15 Kelvin
h = 18 W/(m^2*K)
We need to calculate the current (I) passing through the wire, which depends on the voltage. However, the voltage is not given, so we need more information to proceed with the calculation of the steady-state temperature.