Effect of bypass valve on flow rate and pressure

Question

I have no background in thermodynamics or fluid dynamics, so please bear with me. Refer to the following figure:

A working fluid with a flow rate of $1~kg/s$ enters a heater at a pressure of $1~MPa$ and temperature $150~^\circ C$, which raises its temperatures to $300~^\circ C$. It then enters a turbine, does work, loses pressure and temperature, and exits at $0.5~MPa$ and $200~^\circ C$ at the same flow rate. The bypass valve's purpose is to control the work output of the turbine by bypassing some of the working fluid from the inlet stream directly to the outlet stream. Suppose that we're free to make all the "simplifying assumptions" for this problem. Suppose also that initially the opening fraction of the bypass valve is $0\%$ i.e., the valve is fully closed. My questions are:

1. How would the pressure and flow rate change as the bypass valve opening fraction is varied ($0\%$ means fully closed, $100\%$ means fully opened)? In particular, I'm interested to know them at points A, B, X, Y and E.
2. Would a high pressure stream at Y intercepting the low pressure stream at E raise the pressure and flow rate of the latter? How could that be quantified?
3. Could the valve's opening fraction be linearly related to flow rate and pressure? Meaning, could it be assumed that flow rate and pressure at point B would be halved if the valve was half-open ($50\%$ opening fraction)?
4. What would be the simplest way to model this valve?

Some questions might seem redundant, but I listed them to get a thorough response, for clarity and understanding. Also, all the numbers are arbitrary and any unreported quantities could be assumed. Thank you.

Answer 1

I'll not get too deep, because my hidraulics basis is truly shallow.

• Assuming all your pipes have the same section (possibiliting the same flow rate of 1kg/s in all points).
• Disconsidering the termal effects, because the heated liquid will expand and increase the pressure in the local (not too sure about this part).
• Disconsidering possible negative effects in the case of reverse flow of the fluid.

Your by-pass valve will control the flow (from 0% to 100%). Like on eletrical sistem the fluid will take the easiest path everytime.

1. How would the pressure and flow rate change as the bypass valve opening fraction is varied (0% means fully closed, 100% means fully opened)? In particular, I'm interested to know them at points A, B, X, Y and E.

Your flow will not change if your section not change (here's exactly what the valve does).
The pressure is fairly constant (not considering the termal effects). So, if your entry point is limited at 1MPa your exit point is limited the same (disconsidering the turbine that "steals" pressure in the sistem)

Let's simulate some cases:

1º - Your valve is totally closed (0%). The readings will be:
point A - 1Kg/s 1MPa 150º
point B - 1Kg/s 1MPa 150º
point C - 1Kg/s 1MPa 300º
point D - 1Kg/s 0,5MPa 200º
point E - 1Kg/s 0,5MPa 200º
point X - 1Kg/s 1MPa 150º
point Y - 1Kg/s 0,5MPa 200º

2º - Your valve is half open (50%). The readings will be:
point A - 1Kg/s 1MPa 150º
point B - 0,5Kg/s 1MPa 150º
point C - ~0,5Kg/s 1MPa 300º
point D - ~0,5Kg/s ~0,6MPa 200º
point E - 1Kg/s ~0,6MPa 200º
point X - 0,5Kg/s 1MPa 150º
point Y - 0,5Kg/s ~0,6MPa 200º
Obs.: The aproximate values here need some real calculation for the true value, i've just put some numbers that I think resemble what should be.

3º - Your valve is totally open (100%). The readings will be:
point A - 1Kg/s 1MPa 150º
point B - 0Kg/s (or close to this) 1MPa 150º
point C - 0Kg/s (or close to this) 1MPa 300º
point D - 0Kg/s (or close to this) 1MPa 200º
point E - 1Kg/s 1MPa 200º
point X - 1Kg/s 1MPa 150º
point Y - 1Kg/s 1MPa 200º

2. Would a high pressure stream at Y intercepting the low pressure stream at E raise the pressure and flow rate of the latter? How could that be quantified?

Yes, there will be a pressure change at E, but this is affected more because of the flow from Y, and not because of the pressure. Btw. The flow at E will be exactly the same unless your sistem lose/gain external flow from another point or from a pump.

3. Could the valve's opening fraction be linearly related to flow rate and pressure? Meaning, could it be assumed that flow rate and pressure at point B would be halved if the valve was half-open (50% opening fraction)?

The flow rate will be linearly related at point B (not considering the termal effects). But the pressure will remain the same.

4. What would be the simplest way to model this valve?

I'm not so sure what you want with this question. By my understanding, you're asquing what is the simplest connection method to get a linear response in your turbine.
And my sincere answer in this case will be, "I don't know". In my understanding the point you've chosen is the best point (no personal experience, so take with a grain of salt).

Answer 2

1. I would read into the concept of the discharge coefficient and its relationship to flow, pressure, density, and area http://www.valvias.com/discharge-coefficient.php This concept is complicated by compressible effects.

2. Mass flow coming into the system must equal the mass flow going out. According to your diagram, there are no leak paths between A and E. Therefore, the flow at E will not change based on Y. The pressure will change though because of the different fluids mixing.

3. See the equation in the first link for a relationship between flow and pressure. The relationship is non-linear.

4. Many 1D flow solvers have methods to model valves, but equations such as the ones provided in the link can suffice.