A variable volume cylindrically shaped perfectly stirred mixer through which a s

Question

A variable volume cylindrically shaped perfectly stirred mixer through which a solution stream is being processed. The exit line is located at the bottom of the cylindrical tank and contains a linear resistance R in the form of a valve. The inlet solution concentration C_1 gm/lit. and flow rate q lit/min, are time dependent, that the tank liquid level is also variable. Write the model equations retailing from total and component mass balance. You may assume that solution density does not depend on concentration and the system is working isothermally If the resulting equations are non-linear. try to linearize them.

Explanation / Answer

Consider a tank of cross sectional area A to which is attached to a valve with resiastance R. Consider liquid height is variable as h. Assume q is the inlet volumetric flowrate and qo is outlet flowrate. CI is inlet concentration and Co is outlet concentration.

Assume that volumetric flow rate through a resistance is related to height h of liquid

ie qo = h/R

A varying volumetric flow q of liquid at constant density enter the tank.

Overall mass balance equation is

(Rate of mass flow in) - (Rate of mass flow out) = (Rate of accumulation of mass in tank)

q(t) - qo(t)= d(Ah)/dt

q(t) - qo(t)= d(Ah)/dt

ie

q(t) - qo(t)= d(Ah)/dt

q - h/R = A dh/dt

So overall mass balance

q - h/R = A dh/dt

If we are considering component balance

(Flow rate of component in) - (Flow rate component out) =(Rate of accumulation of component in tank)

qCI - qoCO = d(VCO)/dt

qCI - qoCO = Ad(hCO)/dt

qCI - (h/R)CO = Ad(hCO)/dt

So component balance is

qCI - (h/R)CO = Ad(hCO)/dt

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