How does one calculate the expected leak rate through a hole, and the pressure drop due to that leak? Let's say we have a pressurized horizontal tube that has a hole along the way. What would be the flow rate through that hole, and what would the pressure be downstream of the hole?
Or if you prefer to work with numbers, let's say the following:


*

*Inlet Pressure: 5 barg

*Diameter of main stream: 1 inch

*Initial Main Flow velocity: 1 m/s

*Hole Diameter: 5 mm


Unknowns:


*

*Flow rate through hole

*Flow rate downstream of the hole

*The pressure downstream of the hole


I'm aware that conservation of mass can be used to determine one flow rate based on the other, but I'm not sure how to get the flow rate through the hole, and I'm also not sure how to then calculate the pressure drop, since the Bernoulli equation that I'm familiar with predicts a pressure increase due to the decrease in flow speed.
 A: 
I had to use a hole diameter of 1mm instead of 5mm, or the pressure loss would be too big
A: Edit: Found the solution to this, which is to re-derive the Bernoulli equation from energy conservation. The confusion in the original Bernoulli equation is due to the fact that mass is cancelled out, when in this problem it shouldn't be
A: Bernoulli's equation has to be applied to the flow as a whole. It can't be applied to individual parts of the flow. If you split the flow into two subflows, the total of all the energies of the two subflows will be equal to that of the original flow. Since the tube is horizontal, we can ignore gravitational potential energy. Thus we have that 
$m_0(v_0^2+\frac {p_0}{\rho}) = m_1(v_1^2+\frac {p_1}{\rho})+m_2(v_2^2+\frac {p_2}{\rho})$ 
where $v_0$ is the velocity before the hole, $v_1$ is the velocity in the tube after the hole, and $v_2$ is the velocity of the fluid escaping the hole, likewise for $p_0, p_1, p_2$, and $m_0, m_1, m_2$ are weights representing the relative masses of the flows.
