CJC0117
Chemical
- Mar 4, 2013
- 19
The method used by DIERS for two-phase, homogenous pipe flow is isenthalpic HEM numerical integration. For a constant-area pipe, pressure increments are made from the upstream pressure, P[sub]1[/sub], to the pressure at the exit of the pipe, P[sub]2[/sub] (which may or may not be the choked pressure), and then length increments are calculated for each step. The total maximum equivalent length for a given set of conditions is the sum of all of the length increments.
ΔL = - [ΔP+G[sup]2[/sup]Δν] / [(2fG[sup]2[/sup]ν[sub]avg[/sub]/D)+(gcosθ/ν[sub]avg[/sub])]
L[sub]eq,max[/sub] = ∑ΔL
Where:
ΔL = length increment for a given pressure increment
ΔP = pressure increment
Δν = change in two-phase specific volume for a given pressure increment
f = Fanning friction factor (may be treated as constant for a constant diameter pipe if flow is in fully turbulent zone)
G = mass flux = W/A (where W is mass flow rate and A is pipe area)
ν[sub]avg[/sub] = average two-phase specific volume for a given pressure increment
D = pipe diameter
g = acceleration due to gravity
cosθ = H/L[sub]eq[/sub] according to DIERS (where H is the elevation change between the pipe inlet and outlet, and L[sub]eq[/sub] is the equivalent length of pipe)
If ΔL reaches zero for a given ΔP, then the choked pressure has been reached, and the maximum allowable equivalent pipe length is the sum of all the length increments calculated.
If one knows the actual equivalent pipe length, L[sub]eq,actual[/sub], and is trying to find the maximum mass flow rate, W, then W is guessed until L[sub]eq,actual[/sub] = ∑ΔL.
The inlet pressure to the pipe, P[sub]1[/sub], may be found by treating the entrance to the pipe (from the vessel) as a nozzle, and finding the nozzle throat pressure corresponding to the mass flux at the entrance to the pipe. This is the method recommended by DIERS, especially for long piping systems.
Now, where I'm having trouble is applying this procedure to systems that have bends or inclines (i.e. cosθ = H/L[sub]eq[/sub] varies) or expansions/contractions (i.e. pipe diameter, friction factor, and mass flux vary). I haven't really been able to find any good examples or detailed discussion of this anywhere. I thought of splitting the entire pipe into smaller segments with constant area and incline angle, and then starting at the upstream pressure, P[sub]1[/sub], and working my way down to the exit pressure, P[sub]2[/sub]. So for the first pipe segment right after the entrance to the pipe, I have to calculate the exit pressure for that individual pipe segment that results in ∑ΔL being equal to the equivalent length of that individual pipe segment. Then I would use that exit pressure as the inlet pressure for the next segment, and so on and so forth until I reach a choke point or the end of the entire pipe.
When the last pipe segment is reached, I'm supposed to keep calculating length increments until I reach the choke point (ΔL=0). Then L[sub]eq,total,max[/sub] = L[sub]eq,1st segment[/sub] + L[sub]eq,2nd segment[/sub] + L[sub]eq,3rd segment[/sub] +...+ (∑ΔL)[sub]last segment[/sub]. If L[sub]eq,total,max[/sub] > L[sub]eq,total,actual[/sub] then W was guessed too small and the whole process is repeated. If L[sub]eq,total,max[/sub] < L[sub]eq,total,actual[/sub], then W was guessed too large. If L[sub]eq,total,max[/sub] = L[sub]eq,total,actual[/sub] within a given tolerance, then W was guessed right.
I tried to apply this procedure and where I run into problems is that I guessed W too small because L[sub]eq,total,max[/sub] > L[sub]eq,total,actual[/sub]. So I go back and increase W. But I reach a certain point where I cannot increase W anymore because if I do, then the flow becomes choked at the entrance to the pipe (which I treat as nozzle flow, as mentioned above). Assuming this is not a computational error on my part, then where am I going wrong? How am I supposed to interpret this result physically?
Also, another problem I've experienced is that I'm not really sure how to treat expansions in the above procedure. The method used above is an equivalent length method (i.e. L[sub]eq[/sub] values are used for fittings), not a resistance coefficient method (i.e. K values are used for fittings). And as far as I know, L[sub]eq[/sub] values cannot be calculated from K values for expansions and contractions, because the pipe diameter is changing.
Hopefully this makes sense and isn't too convoluted. I've thought of using the omega method (which bypasses numerical integration by using a two-phase Pν relationship, and for which Leung provides design charts) but I still foresee myself running into the problem where I'm not able to guess a W large enough to satisfy the given conditions, because choking occurs at the entrance.
ΔL = - [ΔP+G[sup]2[/sup]Δν] / [(2fG[sup]2[/sup]ν[sub]avg[/sub]/D)+(gcosθ/ν[sub]avg[/sub])]
L[sub]eq,max[/sub] = ∑ΔL
Where:
ΔL = length increment for a given pressure increment
ΔP = pressure increment
Δν = change in two-phase specific volume for a given pressure increment
f = Fanning friction factor (may be treated as constant for a constant diameter pipe if flow is in fully turbulent zone)
G = mass flux = W/A (where W is mass flow rate and A is pipe area)
ν[sub]avg[/sub] = average two-phase specific volume for a given pressure increment
D = pipe diameter
g = acceleration due to gravity
cosθ = H/L[sub]eq[/sub] according to DIERS (where H is the elevation change between the pipe inlet and outlet, and L[sub]eq[/sub] is the equivalent length of pipe)
If ΔL reaches zero for a given ΔP, then the choked pressure has been reached, and the maximum allowable equivalent pipe length is the sum of all the length increments calculated.
If one knows the actual equivalent pipe length, L[sub]eq,actual[/sub], and is trying to find the maximum mass flow rate, W, then W is guessed until L[sub]eq,actual[/sub] = ∑ΔL.
The inlet pressure to the pipe, P[sub]1[/sub], may be found by treating the entrance to the pipe (from the vessel) as a nozzle, and finding the nozzle throat pressure corresponding to the mass flux at the entrance to the pipe. This is the method recommended by DIERS, especially for long piping systems.
Now, where I'm having trouble is applying this procedure to systems that have bends or inclines (i.e. cosθ = H/L[sub]eq[/sub] varies) or expansions/contractions (i.e. pipe diameter, friction factor, and mass flux vary). I haven't really been able to find any good examples or detailed discussion of this anywhere. I thought of splitting the entire pipe into smaller segments with constant area and incline angle, and then starting at the upstream pressure, P[sub]1[/sub], and working my way down to the exit pressure, P[sub]2[/sub]. So for the first pipe segment right after the entrance to the pipe, I have to calculate the exit pressure for that individual pipe segment that results in ∑ΔL being equal to the equivalent length of that individual pipe segment. Then I would use that exit pressure as the inlet pressure for the next segment, and so on and so forth until I reach a choke point or the end of the entire pipe.
When the last pipe segment is reached, I'm supposed to keep calculating length increments until I reach the choke point (ΔL=0). Then L[sub]eq,total,max[/sub] = L[sub]eq,1st segment[/sub] + L[sub]eq,2nd segment[/sub] + L[sub]eq,3rd segment[/sub] +...+ (∑ΔL)[sub]last segment[/sub]. If L[sub]eq,total,max[/sub] > L[sub]eq,total,actual[/sub] then W was guessed too small and the whole process is repeated. If L[sub]eq,total,max[/sub] < L[sub]eq,total,actual[/sub], then W was guessed too large. If L[sub]eq,total,max[/sub] = L[sub]eq,total,actual[/sub] within a given tolerance, then W was guessed right.
I tried to apply this procedure and where I run into problems is that I guessed W too small because L[sub]eq,total,max[/sub] > L[sub]eq,total,actual[/sub]. So I go back and increase W. But I reach a certain point where I cannot increase W anymore because if I do, then the flow becomes choked at the entrance to the pipe (which I treat as nozzle flow, as mentioned above). Assuming this is not a computational error on my part, then where am I going wrong? How am I supposed to interpret this result physically?
Also, another problem I've experienced is that I'm not really sure how to treat expansions in the above procedure. The method used above is an equivalent length method (i.e. L[sub]eq[/sub] values are used for fittings), not a resistance coefficient method (i.e. K values are used for fittings). And as far as I know, L[sub]eq[/sub] values cannot be calculated from K values for expansions and contractions, because the pipe diameter is changing.
Hopefully this makes sense and isn't too convoluted. I've thought of using the omega method (which bypasses numerical integration by using a two-phase Pν relationship, and for which Leung provides design charts) but I still foresee myself running into the problem where I'm not able to guess a W large enough to satisfy the given conditions, because choking occurs at the entrance.
