Answer to Question #193643 in Mechanical Engineering for josh

Question

The 96 prismatic lithium-ion battery cells were manufactured in Korea and the Scottish firm Axeon assembled them into a 1411-pound, 338-volt pack that provides a 71-kWh energy capacity and a claimed 1D miles of range [Note: replace D with your ID’s last two digits].

 

(d) Using the charge capacity of the battery derived in c, use dimensional analysis techniques to:

1.                 Determine the dimensions of Q using only mass, length and time.

2.                  Determine the equation that governs charging time of the battery.

3.                 If the charging current is 10 A. Find charging time assuming the values provided in c are valid values for part d.

Answer 1

(d)1. Dimension of Q in terms of m ; L&t is $kg^3m^{5/2}$

2) If Q be the charge on capacitor having capacitance C &Resistance R, at any time t, then charging

$E=\frac{q}{c}+R_1$

$=\frac{q}{c}+R \frac{dq}{dt}$

$=\frac{dq}{q-CE}=-\frac{dt}{RC}$

$=\int_0^q \frac{dq}{q-CE}=-\frac{1}{RC}\int_0^t dt$

$=ln(\frac{q-CE}{-CE}=-\frac{t}{RC}$

$=q=CE(1-e^{-t/RC})$

$q=q_0(1-e^{-t/RC})$

Here RC represent the time constant

Hence current

$i=\frac{dq}{dt}=\frac{E}{R} e^{-t/RC}=e^{-t/RC}$