I lost about 10kg.
At the beginning, i was able to burn more calories because of the extra 10kg.
If I carry a backpack of 10kg with me now while using elliptical cycle, will the burn rate be same as before?
Assuming my HR is the same.
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No, it won't.
The 10kg that you lost was living tissue, whether fat or not. It had a metabolic cost to carry around and sustain, whether during cardio or sitting at home. You can carry 10 kg in a vest or something, but it's not going to be the same as having 10kg of body weight.
If you are exchanging fat for building muscle, you are ahead of the game anyway, as muscle consumes more calories than fat on a daily basis anyway.
I would say don't worry about it as your cardio machine is probably innacurate anyway. Yes as you lose weight your metabolic needs go down, but all you have to do is readjust your calorie intake and keep going.
FYI For more accurate calorie tracker is this from Journal of Sports Science http://www.braydenwm.com/cal_vs_hr_ref_paper.pdf Calories Burned = [(Age x 0.074) - (Weight x 0.05741) + (Heart Rate during exercise x 0.4472) - 20.4022] x Time / 4.184
First, the human body's ability to adapt: Calorie burned at the beginning of an exercise program is usually higher than those burned after performing the program for a while (assuming the same exercises and intensity are used). This is because the body was not accustomed to the process and had to work harder to perform the exercises; thus, you struggle with the exercises at the beginning.
As you progress in the program, your body becomes acquainted with the process and becomes better at handling the same tasks (and load). As a result, you no longer struggle (or not as much) with the exercises as previously. The number of calories you burn at the same intensity reduces (even if your body weight remains the same).
Second, backpack is an external load acting on the body. The body handles external and internal loads differently. Because the increase in internal load (weight gain) is gradual, the body adapts to the increment. On the other hand, an external load is drastic in changes. The body will initially struggle with this new load and will progressively adapt. Without external signs (such as breaking a chair after sitting on it, fat deposits popping up everywhere, loose skin, etc), people wouldn't realize they're packing on weight; on the other hand, you'll easily notice if a 20-lb item was dropped on you. Similar process when the weights are dropped.
With the same intensity, I think you should burn more calories initially with a backpack because the body hasn't adjusted to the new load. You should also feel the soreness in your upper body as well. The burn will decrease as the body adapts to the new load.
This is why most weight-loss programs use progressive loads or variations in exercise complexity in order to prevent the body from fully adapting to the process.
Progressively increase the backpack weight or progressively increase your intensity. Your burn rate won't be the same; however, it won't stall as long as your body is being challenged.
Disclaimer: This answer is based on observations, common sense, and my perception on how the human body works. If any of these is directly contradict any scientific-based paper, comment or update the answer.