Situations arise in engineering when we want to control the flow rate of a solution. This solution could be pumped into a holding tank, distillation column, heat exchanger, reactor, etc. This means that the flow rate becomes a function of time. When we turn on a pump, the flow rate is (for the most part) constant. The problem is that we may want the flow rate to be very fast at first, and then very slow towards the end of the process. This is especially true in certain chemical reactors where the concentration of reactants is always important. Control systems that could handle this type of situation are very expensive. Pumps (relatively speaking) are cheap. So, having a bank of pumps will usually solve the problem. The pumps are simply turned on or off to regulate the flow. Mathematically, this amounts to taking the volumetric flow rate function and finding a piecewise-linear approximation. Suppose that, because of the application we are running, we need a flow rate given by the function ,where t is in hours and v(t) is in 100 cubic feet per hour (e.g., v(0)=1 which means that at time t=0, we need a flow rate of 1*100=100 cubic feet per hour). Find a linear approximation and a piecewise-linear approximation using first two lines and then three lines. Think about what using more lines would do to the approximation. You will need to find suitable values of t to be able to find your approximation. As a further exercise, try to find values of t for your approximation that keeps the error (i.e., the difference between the approximation and the real function v(t) ) less than 0.1.Hint: to help with this, ask yourself this question: where would more lines be useful - where the function changes a lot or where the function changes a little?]
Situations arise in engineering when we want to control the flow rate of a solution. This solution could be pumped into a holding tank, distillation column, heat exchanger, reactor, etc. This means that the flow rate becomes a function of time. When we turn on a pump, the flow rate is (for the most part) constant. The problem is that we may want the flow rate to be very fast at first, and then very slow towards the end of the process. This is especially true in certain chemical reactors where the concentration of reactants is always important. Control systems that could handle this type of situation are very expensive. Pumps (relatively speaking) are cheap. So, having a bank of pumps will usually solve the problem. The pumps are simply turned on or off to regulate the flow. Mathematically, this amounts to taking the volumetric flow rate function and finding a piecewise-linear approximation. Suppose that, because of the application we are running, we need a flow rate given by the function ,where t is in hours and v(t) is in 100 cubic feet per hour (e.g., v(0)=1 which means that at time t=0, we need a flow rate of 1*100=100 cubic feet per hour). Find a linear approximation and a piecewise-linear approximation using first two lines and then three lines. Think about what using more lines would do to the approximation. You will need to find suitable values of t to be able to find your approximation. As a further exercise, try to find values of t for your approximation that keeps the error (i.e., the difference between the approximation and the real function v(t) ) less than 0.1.Hint: to help with this, ask yourself this question: where would more lines be useful - where the function changes a lot or where the function changes a little?]
