n our text book the following equation is given:

P=(pm - p)g * R
where P is the pressure drop
R is the manometer reading
pm is the density of the manometer fluid
p is the density of the process fluid in the pipe
g is the acceleration do due to gravity
Then our professor told us this:
It is known that the flow rate measured by the orifice plate / manometer system is proportional to the square-root of the measured pressure difference. If the flow rate of the fluid is 123 gpm (gallons per minute) when the manometer reading is 32 cm, what is the maximum flow rate that can be measured by this manometer?
I have no idea how to solve for this please help.

To solve this problem, you need to understand how to relate the flow rate to the pressure difference and how to manipulate the given equation.

The equation given to you, P = (pm - p)g * R, represents the pressure drop (P) across the orifice plate-manometer system. In this equation, R is the manometer reading and pm is the density of the manometer fluid. p represents the density of the process fluid in the pipe, and g is the acceleration due to gravity.

According to your professor, the flow rate measured by the orifice plate/manometer system is proportional to the square root of the pressure difference. Mathematically, we can represent this as:

Flow rate ∝ √(P)

Let's consider the equation P = (pm - p)g * R. We can rearrange it to solve for P:

P = (pm - p)g * R
P = g * (pm - p) * R

Now, we have the pressure drop (P) in terms of known variables. We can plug this into the proportionality equation for flow rate:

Flow rate ∝ √(g * (pm - p) * R)

The flow rate is given as 123 gpm when the manometer reading (R) is 32 cm. We need to find the maximum flow rate that can be measured by this manometer.

To find the maximum flow rate, we need to determine the maximum pressure difference (P) that the manometer can measure.

Substituting the given values into the proportionality equation:

123 gpm ∝ √(g * (pm - p) * 32 cm)

Now, we need to solve this equation to find the maximum flow rate. Here are the steps:

1. Square both sides of the equation to eliminate the square root:

(123 gpm)^2 = g * (pm - p) * 32 cm

2. Rearrange the equation:

(g * (pm - p) * 32 cm) = (123 gpm)^2

3. Divide both sides by (32 cm) to isolate the term (pm - p):

g * (pm - p) = (123 gpm)^2 / (32 cm)

4. Divide both sides by g:

(pm - p) = (123 gpm)^2 / (32 cm * g)

5. Now, you can calculate the maximum flow rate by plugging in the known values for pm, p, g, and R into the equation:

Maximum flow rate = √[ g * (pm - p) * R ]

= √[ g * ((123 gpm)^2 / (32 cm * g)) * 32 cm]

Simplifying the equation:

Maximum flow rate = √[ (123 gpm)^2 * 32 cm / 32 cm]

= 123 gpm

Therefore, the maximum flow rate that can be measured by this manometer is 123 gpm.