19 Mar '17 15:531 edit
One of our process at the mill I currently work at is electro patenting. It is done in line ahead of a electro plating process. The wire comes in contact with a voltage source and ground as it is passing through the line causing it to heat up, which changes it microstructure etc.. I'd like to try to find the model for the temperature as a function of position between the electrical contacts. Neglecting (for now) heat transfer by radiation. I would like to try to take into account axial conduction, but because of my inexperience with using differentials, Im not sure I can do what I'm doing and am looking for some confirmation.
Start with a differential volume of the wire as it passes between contacts, and perform a power balance.
Cross Sectional Area = A
Length = dL
E'_stored = E'_in - E'_out (Eq1)
E'_stored = m' * c * dT
where,
m' = mass flow rate of steel wire
c = specific heat of steel
dT = differential change in temperature across the element
Next,
E'_in = V * I + k * A * dT/dL
where,
V = Voltage
I = Current
k = Thermal Conductivity of Steel
dT/dL = Thermal gradient across the element
Then,
E'_out = ( V - dV ) * I + dq_h
E'_out = ( V - dV ) * I + h* π * D * ( T - T_∞ ) * dL
where,
h = convection coefficient
T = Temperature as a function of position ( "L" )
D = diameter of wire
T_∞ = Ambient Temperature
Plugging all this back into Eq1, and simplifying:
m' * c * dT = I * dV - h* π * D * ( T - T_∞ ) * dL + k * A * dT/dL
dV is the voltage drop across the element
which is given by:
dV = I * dR = I * p * dL / A
where,
I = Current
p = coefficient of resistivity for steel ( assumed constant)
A = crossectional area of wire
Subbing that relationship in:
m' * c * dT = I² * p / A * dL - h* π * D * ( T - T_∞ ) * dL + k * A * dT/dL
Here is the part where I'm unconfident ( If i should be even less confident at this point let me know ). Divide through by the differential dL and re-arrange into standard form to give a second order linear ODE.
-k * A * d²T/dL² + m' * c * dT/dL + h* π * D * ( T - T_∞ ) = I² * p / A
Is the term for axial conduction I derrived here in bold legitimate?
Start with a differential volume of the wire as it passes between contacts, and perform a power balance.
Cross Sectional Area = A
Length = dL
E'_stored = E'_in - E'_out (Eq1)
E'_stored = m' * c * dT
where,
m' = mass flow rate of steel wire
c = specific heat of steel
dT = differential change in temperature across the element
Next,
E'_in = V * I + k * A * dT/dL
where,
V = Voltage
I = Current
k = Thermal Conductivity of Steel
dT/dL = Thermal gradient across the element
Then,
E'_out = ( V - dV ) * I + dq_h
E'_out = ( V - dV ) * I + h* π * D * ( T - T_∞ ) * dL
where,
h = convection coefficient
T = Temperature as a function of position ( "L" )
D = diameter of wire
T_∞ = Ambient Temperature
Plugging all this back into Eq1, and simplifying:
m' * c * dT = I * dV - h* π * D * ( T - T_∞ ) * dL + k * A * dT/dL
dV is the voltage drop across the element
which is given by:
dV = I * dR = I * p * dL / A
where,
I = Current
p = coefficient of resistivity for steel ( assumed constant)
A = crossectional area of wire
Subbing that relationship in:
m' * c * dT = I² * p / A * dL - h* π * D * ( T - T_∞ ) * dL + k * A * dT/dL
Here is the part where I'm unconfident ( If i should be even less confident at this point let me know ). Divide through by the differential dL and re-arrange into standard form to give a second order linear ODE.
-k * A * d²T/dL² + m' * c * dT/dL + h* π * D * ( T - T_∞ ) = I² * p / A
Is the term for axial conduction I derrived here in bold legitimate?