03_PressureFlowTheory

Pressure Flow Theory 1

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Pressure Flow Theory

© 2004 Aspen Technology. - All Rights Reserved.

2 Pressure Flow Theory

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WorkshopThe Pressure Flow Theory module introduces you to the underlying concepts necessary for developing your own dynamic simulations with HYSYS Dynamics. Some of the things you will learn from this module are:

• The underlying assumptions of dynamic simulation with HYSYS• How to analyze your Flowsheet to make appropriate pressure

flow specifications• Which pressure-flow specifications make sense• How to troubleshoot the process Flowsheet for inconsistent

pressure-flow specifications

Learning ObjectivesOnce you have completed this section, you will understand:

• The basic concepts of dynamic simulation in HYSYS• Dynamic pressure flow specifications• Process Flowsheets


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Theoretical Foundations

The Pressure-Flow Solver: A Boundary Value ProblemIn terms of pressures and flows, perhaps the simplest way to view the pressure flow solver in HYSYS Dynamics is to consider the Flowsheet as a Boundary Value Problem.

If you were to make pressure or flow specifications on all the boundary streams (feeds/product streams in a Flowsheet), then all the internal pressures and flows would be solved simultaneously at each integration step by the pressure-flow solver. The internal stream pressures and flow rates are calculated from the pressure gradients in the Flowsheet. Flow rates are determined from:

1. Changes in vapour pressure nodes (vessels with hold-ups) within the Flowsheet system.

2. Resistance across valves

3. Conductance through equipment (coolers, heaters, heat exchangers)

Pressure NodesAll unit operations (with hold-up) represent pressure nodes. Some unit operations may contribute to one or more nodes. For example:

• Heaters/Coolers with multiple zones• Heat Exchanger - shell side/tube side• Columns with multiple stages (trays)

Since pressure gradients are the driving force for flow in HYSYS, care should be taken to ensure that the pressure profile of the flowsheet has been properly specified.


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Fundamental Principle

Vessel equipment has a fixed geometry and thus a fixed volume. Mathematically, this means that:

Therefore, for a fixed volume, a pressure node (vessel pressure) is calculated as a function of the vessel temperature and the vessel hold up.

In dynamic mode, the rate of change in vessel pressure is related to the rate of change of temperature (enthalpy) and the rate of change of material hold-up (level):

where: V = Fixed volume

F = Change in flow (hold-up)

T = Temperature (change in enthalpy)

A volumetric flow balance around the vessel can be expressed as follows:

where: ∆Vp = Volume change due to pressure change

∆VF = Volume change due to flow changes

∆VT = Volume change due to temperature change

The total volume change must always be zero.

(1)

(2)

(3)

This concept is fundamental to performing dynamic simulation analyses with HYSYS. dV

dt------- 0=

dPdt------- fn V F T, ,( )=

∆VP ∆VF ∆VT 0=+ +


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Example

Consider the operation of a separator in dynamic mode that is initially at steady state with a level of 60%:

Remember: In Steady State,

Flow into separator = Flow out of separator,

no accumulation.

But in Dynamics, if the separator feed flow increases with the product flow rates (vapour and liquid) remaining unchanged, the level (hold up), temperature (enthalpy) and pressure of the vessel must all change from the steady state condition.

Liquid Level Increases

Since

Liquid Flow In - Liquid Flow Out = Accumulation (hold-up),

an increase in the feed liquid Flowrate with a constant liquid product Flowrate results in the liquid level (hold-up) increasing.

Figure 1

Fixed geometry

Assume fixed flow

Assume fixed flow

Flow in

60%


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Vessel Pressure Increases

The vessel pressure would increase for two reasons.

1. Vapour Flow In - Vapour Flow Out = Accumulation.

An increase in the feed vapour Flowrate with a constant vapour product Flowrate results in the vapour (hold up) increasing. Because vapour is a compressible fluid, the accumulation of vapour, occupying a smaller volume, will cause the vessel pressure to rise.

2. The increase in liquid level also causes the vapour hold-up to occupy a smaller volume within the vessel, causing the vessel pressure to rise.

Distributed and Lumped ModelsMost chemical engineering systems have thermal and component gradients in three dimensions (x, y, z) as well as in time. This is known as a distributed system. Thus, in the formulation of chemical engineering problem equations, we obtain a set of partial differential equations in the x, y, z and t domains.

If the x, y, and z gradients are ignored, the system is lumped and all the physical properties are considered to be equal in space. In such, an analysis in which only the time gradients are considered, the chemical engineering system equations are represented by a set of ordinary differential equations (ODE's). This method saves calculation time and provides a solution that is reasonably close to the distributed model solution.

HYSYS uses lumped models for all unit operations. For instance, in the development of the equations describing the separator, it is assumed that there are no thermal, pressure or concentration gradients present. In other words, the temperature, pressure, and component gradients are the same throughout the entire separator.

HYSYS does take into account the static pressures in the fluid and vapour phases. This can result in a dP/dz effect in a vessel. However, HYSYS does not solve any partial differential equations.


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Pressure-Flow Relationship for ValvesIn any Flowsheet, the valve unit operation describes the resistance to flow between two material streams by the Turbulent Flow equation:

where: P1 = upstream pressure (pressure of stream 1)

P2 = downstream pressure (pressure of stream 2)

Cv = the valve coefficient, HYSYS will calculate this value on request

Pressure-Flow Relationship for Other OperationsMore generally, flow rates in HYSYS Dynamics are related to delta P. All process equipment relates the flow between its feed and product streams with flow equations that are similar to the turbulent flow equation. The form of these equations is:

where: k = Conductance, which is a constant representing the reciprocal of resistance to flow

ρ = Stream bulk density

∆P = Pressure gradient across the operation

Specifying Cv or k values, rather than a fixed delta P, across valves and process equipment provides for a more realistic simulation. By specifying these variables, the pressure drop through valves and process equipment can change with changes in flow, as would happen in an actual plant. This allows the Dynamic simulator to more accurately model the actual operating conditions of the plant.

(4)

(5)

FValve fn Cv P1 P2, ,( )=

F k ρ P∆=


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Pressure/Flow NetworksIn HYSYS Dynamics the pressure/flow network is described in terms of nodes, resistance and conductance. Flow takes place in streams from one node to another. Thus there are two basic sets of equations that define the pressure/flow network:

1. Equations that define the material balance at the nodes

2. Equations that define the flow - conductance and resistance to flow

The simplest case is that of incompressible flow with no accumulation at the nodes. In this situation, the flow equations are a function of the pressure gradient and equipment parameters such as the pipe diameter and roughness. The material balance at the nodes is simply that the accumulation is zero.

In a more comprehensive dynamic simulation the pressure flow equations are more complex. They account for:

• Multi-phase flow with the potential for slippage between phases• The rate of change of pressure at the nodes as a function of the

equipment geometry, hold-up and enthalpy of the phases• Flow rates that are determined not only by pressure gradient but

also by weir heights (columns) and density differences

Simultaneous Solution Approach to Pressure Flow Balances

Since pressures at nodes are a function of the flow rates into and out of the nodes and the flow rates through equipment are functions of the upstream and downstream pressures, the relationships between pressure and Flowrate equations in HYSYS Dynamics are significantly coupled. To find a solution to the pressure-flow relationships in HYSYS Dynamics, a simultaneous solution of the Flowsheet is performed. Solving for the flows and pressures requires the simultaneous solution of a set of linear and non-linear equations.

The resistance to flow through valves and the conductance through process equipment determines stream flow rates between nodes.


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• P1, P2, P3 etc. represent Pressure Nodes (Vessels with hold up)• F1, F2, F3 etc. represent streams with flow rates

Moreover, in order to epitomize computational effort, HYSYS Dynamics partitions the equations describing any unit operation into three classes:

• Pressure/flow relationships• Energy relationships• Compositional relationships

These groups of equations can then be integrated/solved with different frequencies. Typically, the pressure flow relationships will have the smallest step size and the composition relationships the largest.

The grouping of the equations also permits a different solution strategy to be applied to each group. In particular, it is possible to solve the pressure/flow relationships simultaneously across the entire Flowsheet while the other equations (composition, enthalpy) are solved on a module by module basis.

Figure 2

P1 P2

P3

P4 P5F1 F4 F5

F3F2


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If you suspect that the P/F solver is failing because of the interaction with the VLE correlation, then you can do one of the following:

• Reduce the integration step size - this can be accessed from the menu bar: Simulation - Integrator.

• Change the frequency of integration steps per step size (composition and enthalpy). This can be accessed from the menu bar: Simulation - Integrator - Execution.

Figure 3


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Degrees of Freedom AnalysisIn Module 2 we introduced the concept of dynamic specifications. The simultaneous solution of the pressure-flow relationships within the Flowsheet requires the user to make a number of dynamic operating specifications.

• P = Pressure• F = Flow

In this Flowsheet, there are 7 variables in total that will define the system. These are:

1. Feed1 (pressure, flowrate - 2 variables)

2. Product1 (pressure, flowrate - 2 variables)

3. Product2 (pressure, flowrate - 2 variables) and

4. V-100 (Pressure - 1 variable)

In addition, there are 4 equations that define the pressure-flow relationships in the Flowsheet. These are:

1. VLV-100: Resistance to Flow equation FVLV-100 = fn(CV, P1, P2)

2. VLV-101: Resistance to Flow equation FVLV-101 = fn(CV,P1, P2)

3. VLV-102: Resistance to Flow equation FVLV-102 = fn(CV, P1, P2)

4. V-100: Pressure Node Relationship dP/dt = fn(V,F,T)

With 7 variables and 4 equations, the DOF = 7-4 = 3. Therefore, 3 P/F specifications need to be made to define this system.

Figure 4


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Understanding the Placement of P/F SpecificationsWhy do some P/F specifications work while others don't?

HYSYS Dynamics is equipped with a Dynamics Assistant that analyzes the Flowsheet to identify problems. (We will discuss this simulation aid later in this module). However, with a greater understanding of the role of the P/F solver and the P/F calculations you will be better able to:

• Specify the process Flowsheet correctly• Troubleshoot the process Flowsheet to identify P/F problems• Use the power of HYSYS Dynamics to its full capabilities

Making Consistent Pressure or Flow Specifications

As mentioned earlier, HYSYS Dynamics users can select from a variety of pressure-flow specification combinations to solve the process Flowsheet. These include:

• Pressure specifications on material streams• Flow specifications on material streams• Fixed pressure drop specifications across equipment• Pressure/Flow calculations for valves - resistance to flow (Cv)• Conductance calculations (k) for process equipment

In the previous example, we had three Degrees of Freedom, requiring that three specifications be made to define the system.


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One possible solution

Specify:

• Feed 1 Pressure• Stream 1 Pressure• VLV-100 Delta P

Although making these 3 specifications will satisfy the DOF analysis, the choice of specifications would not make sense. PFeed1, P1 and PVLV-100 are all related by the following equation:

Specifying the Flowsheet in this manner would lead to an inconsistent solution. In fact the Flowsheet would be under-specified because one of the specifications is redundant.

Figure 5

(6)PFeed1 P1– PVLV 100– 0=∆–


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Another possible solution

Specify:

• Feed1 Pressure• Product1 Pressure• Product2 Pressure

Consider the same Flowsheet with pressure specifications made on all the boundary streams. This solution is consistent because the pressure in the vessel is calculated by the hold-up equation. (The stream flow rates were calculated using the turbulent equation or the resistance to flow equation).

Guidelines to Remember:

Figure 6

• One P/F specification should be made on each boundary stream (feeds/products)

• Make pressure specifications on boundary streams attached to process equipment that use resistance to flow/conductance relationships.

• HYSYS Dynamics will use the equipment conductance or Cv value combined with the pressure of the inlet and outlet streams to determine a Flowrate through the equipment.

• Internal flow rates will be calculated by pressure gradients (resistance to flow/conductance equations) throughout the Flowsheet.


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Summary of P/F Theory and Specifications:1. The flow through the plant, or operation, is driven by the pressure

gradient.

2. P/F theory defines the relationship between flow and pressure.

3. The HYSYS P/F solver solves a set of linear and non-linear equations simultaneously to determine the P/F relationship.

4. In order for the P/F solver to solve the Flowsheet, there must be a pressure gradient established over the entire Flowsheet.

5. The pressure gradient exists due to a specified pressure flow relationship (or a specified pressure drop) over all operations in the Flowsheet.

6. The P/F solver works by finding P from F, according to the P/F theory, or by solving the pressure node equation.

7. Following any flow path through the Flowsheet, the user should be able to see the pressure gradient, or expect to see a pressure gradient established along the path. If the pressure gradient can not be seen, an additional pressure specification may be needed.

8. If the user follows a flow path to the boundary of the Flowsheet, they should see that at such a location, a pressure gradient does not exist, nor can it be established. This means that an pressure (or flow) specification is always needed on boundary streams.

Other possible solutionsIf we modelled the same unit operation without using valves on all product streams, then we could not make P specifications on all boundary streams. Remember the lumped parameter model - the model assumes there are no pressure gradients inside the unit operation. Thus if a pressure specification is made on the vapour product stream it is best not to make pressure specifications on the other unit operation streams. This can lead to an inconsistent solution because once one stream pressure is known they all become known, resulting in no pressure gradients in the unit operation.


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It is possible to have flow specifications on all unit operation streams as long as the vessel pressure is controlled.

Figure 7

Figure 8

Exercise 17

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ExerciseModifying the Flowsheet in dynamics

Load the saved case FHT_Dyn.hsc.

Modify the Flowsheet such that there is a Cooler downstream of the Sep Vap 1 stream.

1. Add a Cooler and move the P/F specifications.

2. Calculate the resistance to flow for the Cooler. Like the Valve, the flow through the Cooler is calculated as a function of delta P.

Using the conductance equation provides a more realistic simulation. The pressure drop across the Cooler can change with increases or decreases in flow. If the Cooler pressure drop were fixed, then it would not vary. Because we specify the resistance to flow as the dynamic specification, then we can make a pressure specification as the boundary specification. The flow rates are calculated by the resistance equation.

Save your case as Exercise.hsc.

Save your case!

18 Exercise

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