To calculate the new volume flow rate when the pressure difference is 5.5 times the original pressure, we can use the concept of Bernoulli's equation. Bernoulli's equation states that for an incompressible fluid flowing through a tube, the total energy per unit volume (pressure energy + kinetic energy + gravitational potential energy) is constant along a streamline.
The equation for volume flow rate (Q) in a tube is given by:
Q = A * v
Where:
Q = Volume flow rate (cc/min)
A = Cross-sectional area of the tube (cc)
v = Velocity of the fluid (cm/min)
Since we are assuming the fluid is incompressible, we can use Bernoulli's equation in the following form:
P1 + 0.5 * ρ * v1^2 + ρ * g * h1 = P2 + 0.5 * ρ * v2^2 + ρ * g * h2
Where:
P1 = Original pressure (dynes/cm^2)
v1 = Original velocity of the fluid (cm/min)
ρ = Density of the fluid (g/cc)
g = Acceleration due to gravity (cm/s^2)
h1 = Original height of the fluid (cm)
P2 = New pressure (5.5 times the original pressure)
v2 = New velocity of the fluid (to be calculated)
h2 = New height of the fluid (assumed to be the same as the original height, so h2 = h1)
The density of the fluid (ρ) cancels out from both sides of the equation, and since the height of the fluid remains the same, we can simplify the equation to:
P1 + 0.5 * v1^2 = P2 + 0.5 * v2^2
Now, we are given the original volume flow rate (Q1 = 25 cc/min). We can write:
Q1 = A * v1
Similarly, for the new volume flow rate (Q2), we can write:
Q2 = A * v2
Since the tube's cross-sectional area (A) remains constant, we can divide the equations to find the ratio of the new volume flow rate to the original volume flow rate:
Q2 / Q1 = (A * v2) / (A * v1) = v2 / v1
Now, let's substitute the original volume flow rate (Q1 = 25 cc/min) and the given pressure difference (P2 = 5.5 * P1) into the Bernoulli's equation:
P1 + 0.5 * v1^2 = 5.5 * P1 + 0.5 * v2^2
Rearrange the equation to solve for v2:
0.5 * v2^2 = 5.5 * P1 + 0.5 * v1^2 - P1
0.5 * v2^2 = 4.5 * P1 + 0.5 * v1^2
v2^2 = 9 * P1 + v1^2
v2 = sqrt(9 * P1 + v1^2)
Now, we need to find the value of P1. We are only given the volume flow rate (Q1 = 25 cc/min), and we cannot directly calculate the original pressure without additional information like the tube's dimensions and fluid properties.
Therefore, without knowing the original pressure (P1) or any other relevant data, we cannot calculate the new volume flow rate (Q2). The problem is currently incomplete, and more information is needed to perform the calculation.