pbuk said:
Perhaps, but I'm not sure that my logic in post number 62 holds water? I'm not sure that I can assume the temperature of the strip would be 60 deg C, while the temperature of the cell is 51 deg C? Because this would result in heat transfer occurring between the strip and cell. If we want the cells to be the limitation, than we want the strip to be a colder temperature than the cell. So I have to find a way to find the resistance of the strip, such that heat transfer occurs between the cell and strip, as opposed to the other way around.
Like you had mentioned, I should consider the cells being the limitation factor, and not the strips. Meaning that the cells should be hotter than the strips.
Problem Statement:
Determine the dimensions of a nickel strip required to maintain ##30 A##, such that the randomly selected 21700 cells, Samsung 50S, within a battery box. Size the strip such that the cells are the limitation, and not the strip. Meaning that the cells maintain a higher temperature than the strip. Assume that cells are identical, with identical internal resistances ##r_{int}##. Each individual cell is at the same temperature ##T_{C}## when drawing the same current, therefore there is no heat transfer in between each cell. Assume that the ambient air conditions ##T_{\infty} = 100\;deg\;F##. 21700 cells have a diameter of ##21 mm##. Assume that insulation is placed around the nickel strip, such that it is in contact with the side walls of the battery box. This assumption limits convection of the stirp to near zero, and can be considered insignificant. Test data shows that the temperature of this particular cell model is ##51\;deg\;C## at ##10 A##. Find a generic formula with variables.
Heat transfer from cells to strip:
$$(\frac{I}{3})^{2}\;r_{int} + (\frac{I}{3})^{2}\;r_{int} + (\frac{I}{3})^{2}\;r_{int} = \frac{k_{S}}{t_{S}}A_{C}(T_{C} - T_{S}) + \frac{k_{S}}{t_{S}}A_{C}(T_{C} - T_{S}) + \frac{k_{S}}{t_{S}}A_{C}(T_{C} - T_{S})$$
$$3(\frac{I^{2}}{9})r_{int} =3\frac{k_{S}}{t_{S}}A_{C}(T_{C} - T_{S})$$
$$\frac{1}{3}I^{2}r_{int} =3\frac{k_{S}}{t_{S}}A_{C}(T_{C} - T_{S})$$
Solving for ##T_{C} - T_{S}##:
$$T_{C} - T_{S} = \frac{1}{3}I^{2}r_{int}\frac{1}{3}\frac{t_{S}}{k_{S}}\frac{1}{A_{C}}$$
$$T_{C} - T_{S} = \frac{I^{2}\;r_{int}\;t_{S}}{9k_{S}\;A_{C}}$$
Heat transfer from strip to insulation:
$$\frac{1}{3}I^{2}r_{int} + I^{2}R = \frac{k_{I}}{t_{I}}A_{S}(T_{S} - T_{I})$$
$$I^{2}(\frac{1}{3}r_{int} + R) = \frac{k_{I}}{t_{I}}A_{S}(T_{S} - T_{I})$$
Solving for ##T_{S} - T_{I}##:
$$T_{S} - T_{I} = (\frac{1}{3}r_{int} + R)\frac{t_{I}I^{2}}{k_{I}A_{S}}$$
Heat transfer from insulation to box:
$$I^{2}(\frac{1}{3}r_{int} + R) = \frac{k_{B}}{t_{B}}A_{I}(T_{I} - T_{B})$$
Solving for ##T_{I} - T_{B}##:
$$T_{I} - T_{B} = (\frac{1}{3}r_{int} + R)\frac{t_{B}I^{2}}{k_{B}A_{I}}$$
Heat transfer from box to ambient air:
$$I^{2}(\frac{1}{3}r_{int} + R) =(T_{B} - T_{\infty})\sum_{i=1}^n h_{i}*A_{i}$$
Solve for ##T_{B} - T_{\infty}##:
$$(T_{B} - T_{\infty}) = \frac{I^{2}(\frac{1}{3}r_{int} + R)}{\sum_{i=1}^n h_{i}*A_{i}}$$
Solve for ##T_{C} - T_{\infty}##:
$$T_{C} - T_{\infty} = T_{C} - T_{S} + T_{S} - T_{I} + T_{I} - T_{B} + T_{B} - T_{\infty}$$
$$T_{C} - T_{\infty} = \frac{I^{2}\;r_{int}\;t_{S}}{9k_{S}\;A_{C}} + (\frac{1}{3}r_{int} + R)\frac{t_{I}I^{2}}{k_{I}A_{S}} + (\frac{1}{3}r_{int} + R)\frac{t_{B}I^{2}}{k_{B}A_{I}} + \frac{I^{2}(\frac{1}{3}r_{int} + R)}{\sum_{i=1}^n h_{i}*A_{i}}$$
$$T_{C} - T_{\infty} = I^{2}(\frac{\;r_{int}\;t_{S}}{9k_{S}\;A_{C}} + (\frac{1}{3}r_{int} + R)(\frac{t_{I}}{k_{I}A_{S}} + \frac{t_{B}}{k_{B}A_{I}} + \frac{1}{\sum_{i=1}^n h_{i}*A_{i}}))$$
Now solve for ##R##:
$$\frac{T_{C} - T_{\infty}}{I^{2}} = \frac{\;r_{int}\;t_{S}}{9k_{S}\;A_{C}} + (\frac{1}{3}r_{int} + R)(\frac{t_{I}}{k_{I}A_{S}} + \frac{t_{B}}{k_{B}A_{I}} + \frac{1}{\sum_{i=1}^n h_{i}*A_{i}})$$
$$\frac{T_{C} - T_{\infty}}{I^{2}} - \frac{\;r_{int}\;t_{S}}{9k_{S}\;A_{C}} = (\frac{1}{3}r_{int} + R)(\frac{t_{I}}{k_{I}A_{S}} + \frac{t_{B}}{k_{B}A_{I}} + \frac{1}{\sum_{i=1}^n h_{i}*A_{i}})$$
$$\frac{\frac{T_{C} - T_{\infty}}{I^{2}} - \frac{\;r_{int}\;t_{S}}{9k_{S}\;A_{C}}}{\frac{t_{I}}{k_{I}A_{S}} + \frac{t_{B}}{k_{B}A_{I}} + \frac{1}{\sum_{i=1}^n h_{i}*A_{i}}} = \frac{1}{3}r_{int} + R$$
$$R = \frac{\frac{T_{C} - T_{\infty}}{I^{2}} - \frac{\;r_{int}\;t_{S}}{9k_{S}\;A_{C}}}{\frac{t_{I}}{k_{I}A_{S}} + \frac{t_{B}}{k_{B}A_{I}} + \frac{1}{\sum_{i=1}^n h_{i}*A_{i}}} - \frac{1}{3}r_{int}$$
Where:
##R## is the resistance of the strip
##T_{C}## is the temperature of the conductor
##T_{\infty}## is the temperature of the ambient air conditions
##I## is the current going through the nickel strip
##r_{int}## is the internal resistance of the cell
##t_{S}## is the thickness of the nickel strip
##k_{S}## is the thermal conductivity of the nickel strip
##A_{C}## is the surface area that the cell is in contact with the strip
##t_{I}## is the thickness of the insulation
##k_{I}## is the thermal conductivity of the insulation
##A_{S}## is the surface area that the strip is in contact with the insulation
##t_{B}## is the thickness of the box wall
##k_{B}## is the thermal conductivity of the box
##A_{I}## is the surface area that the insulation is contact with the box
##h_{i}## is the heat transfer coefficient for surface area ##i## for the box to the ambient air
##A_{i}## is the surface area ##i## of the box in contact with ambient air
Now that I know ##R## I can select an appropriately dimensioned strip that has this resistance.
Does this look better? I'm not making any assumptions about the temperature of the strip ##T_{S}##, other than that it is less than the temperature of the cell ##T_{C}##, which is what, to indicate that the cells are the limitation and not the strip. I don't need to find ##T_{S}## in this method. I'm not sure if my heat transfer for the strip to insulation is correct?
Thanks for all the help! I'm ok accepting that "any" dimensioned strip will work, but I just want to understand why it's insignificant. If it is insignificant, what is the exact point in which it does become significant? Certainly there is a boundary point where I most consider the dimensions. I wonder what this point is and how to find it out.
