My project needs to power a washer pump, PIR and arduino from a rechargeable NiMH battery.
According to the spec sheet for the pump it is 12v with a 17.8W output at maximum efficiency. The pump is motion activated by a PIR sensor so only needs intermittent use, not constant, so to keep the size and cost of the battery down I'm aiming at 2 hours use. So (17.8 x 2)/12 = 2.96AH, right?
But presumably I need to add power for the Arduino board. The Duemilanove specs say it has a 5v operating voltage and 40ma per I/O pin. I'm using 2 pins, one to the PIR and one to the pump. Do I simply add those figures to the battery voltage and maH or is it more complicated than that?
Your calculation for the # of AH for the pump is basically correct. A battery is rated for the number of amp hours it can provide at a certain draw. It may be more or less depending on how fast you discharge it. But your calculation will be as accurate as you need.
You need to get 5V to the arduino somehow. This can be done either through it's onboard LM7805 or through a DC-DC convertor. The DC-DC convertor will be more energy efficient. Now, assuming your arduino needs, say, 200ma then with the LM7805 you'd draw 200ma/H from the battery. The DC-DC convertor would draw more along the lines of about 90ma. Neither one will change the required # of AH much.
As for figuring the power needed for the arduino: For an accurate figure you need to look at the massive reference manual for the ATMEGA168 chip to find out how much power it uses at 16MHz, 5V, and full processing load. It takes power to run the serial UART unless you turn that off, etc, etc. The manual explains all of this. Then, you will certainly need a transistor of some sort to run the pump. I'd assume that unless you are using a MOSFET transistor that you'll need to peak out the 40ma on the pin. I don't know how much draw a PIR takes. It may or may not have to be powered from something other than an arduino pin.
As you can see, getting an accurate calculation could be time consuming. I think you would be pretty safe in thinking that the total power usage of an arduino would not exceed 200-250ma max. But don't count on that unless you really don't want to read the spec sheet for the processor.