Please help me with problem explained below;

Question

Hello, i work in a processing plant, one of our process here involves forwarding of carbon cranules from one tank to the other, the total carbon granules in tonnes in each tank can be determined. However, the required carbon (tonnes) from a tank to be trasfered to the next tank is not actually acurate. Carbon is forwarded by fixed speed carbon forwarding pumps. The carbon is forwarded with slurry so the slurry flow dos'nt decrease except the carbon content in the flow.  The Idea is the carbon (tonnes) in the tank is gradually reduced as the pump runs. Can some one help me formulate a limiting formular (show calculations) for the carbon in the tanks with respect to time. Say the pump flow rate is 30m3/hr, initial carbon tonnes in tank is 10tonnes, required carbon to be trasfered is 2tonnes. How long will it take to achieve 8tonnes in the tank given the tank level will be the same all throughout. I hope im clear enough, for any more clarification on my question, dont hesitate to ask. Thanks

Answer 1

Let R=flow rate provided by the pump in cubic metres per hour. Let T=time, then after a time T hours the volume, V, of carbon transferred is RT cubic metres. One metric tonne is 1000kg. Let the density of the carbon granules be D=W/V where W is the weight in kilogram and V=volume in cubic metres.

So W=VD=RTD.

The amount of carbon, C, left in the forwarding tank after T hours is C=C0-RTD or RTD=C0-C, where C0 is the initial amount in kilogram in the tank. In the example, C0=10000 and C=8000kg or C0-C=2000kg.

We need D in kg/m^3. D is an average density because there is space between the granules, they may also be porous, and there may be other substances mixed with the carbon. For the sake of finding a reasonable answer let D=450kg/m^3 (0.45g/cc). R=30 m^3/hour.

From the formula T=(C0-C)/RD=2000/(30*450)=4/27 hour=8.9 mins approx.