The internal resistances in a ULN2004 stage add up to 20.9 K. For the Thevenin equivalent impedance of a voltage divider, this is in parallel with the external 10 K resistor, for a value of 6764 ohms. With a 470 uF capacitor, one time constant equals 3.81 seconds.
Why one time constant? The voltage on a capacitor charging up from 0 V to some source voltage Vs through a resistance equals 63.2% of Vs after one time constant. With a 10 V source, a 1 M resistor, and a 1 uF capacitor, the time constant is 1 second, and the capacitor voltage will be 6.32 volts after 1 second. The input transition voltage of a 2004 is approx. 5 V. In a 12 V circuit, this is 62.5% of Vs, which is very close to the 1 time constant value. This is the assumption I would use to find starting point values for the timing components.
That 5 V transition level sets a minimum value for the external resistor. At 20 K the input voltage never will go high enough to cause the stage to change state, and that 5 V transition voltage is not tightly defined. If you want a 2:1 adjustment range, then let 10K be the maximum value for the external resistance, calculate the capacitor value needed for the max desired time delay, and then use that capacitor value to calculate the resistance value for the minimum desired time delay.
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