The flow rate of a process stream has tended to fluctuate considerably, creating problems in the process unit to which the stream is flowing. A horizontal surge drum has been inserted in the line to maintain a constant downstream flow rate even when the upstream flow rate varies. A cross section of the drum, which has length L and radius r, is shown below.

The level of liquid in the drum is h, and the expression for liquid volume in the drum is

Here is how the drum works. The rate of drainage of a liquid from a container varies with the height of the liquid in the container: the greater the height, the faster the drainage rate. The drum is initially charged with enough liquid so that when the input rate has its desired value, the liquid level is such that the drainage rate from the drum has the same value. A sensor in the drum sends a signal proportional to the liquid level to a control valve in the downstream line. If the input flow rate increases, the liquid level starts to rise; the control valve detects the rise from the transmitted signal and opens to increase the drainage rate, stopping when the level comes back down to its set-point value. Similarly, if the input flow rate drops, the control valve closes enough to bring the level back up to its set point.

(a) The drum is to be charged initially with benzene (density = 0:879 g/cm³) at a constant rate ṁ = (kg/min) until the tank is half full. If L = 5 m, r = 1 m, and ṁ = 10 kg/min, howlong should it take to reach that point?

(b) Now suppose the flow rate into the tank is unknown. A sight gauge on the tank allows determination of the liquid level, and instructions are to stop the flow when the tank contains 3000 kg. At what value of h should this be done?

(c) After the tank has been charged, the flow rate into the drum, ṁ₁, varies with upstream operations, and the flow rate out is 10 kg/min. Write a mass balance around the drum so that you obtain a relationship between m_ 1 and the rate of change in the height of liquid in the tank (dh/ dt) as a function of h. Estimate the flow rate into the tank when h has an approximate value of 50 cm, and dh/dt = 1 cm/min. Although an analytical solution is feasible, you may find it easier to create plots of V and dV/dh at 0.1 m increments in h, which can be used in obtaining an approximate solution to the problem.

(d) Speculate on why the drum would provide better performance than feeding a signal proportional to the flow rate directly to the control valve that would cause the valve to close if the flow rate drops below the set point and to open if the flow rate rises above that point.

Transcribed Image Text:

V = L - (r - ア-(r-h° cos