## Steady-State Initialization of Conveyors

Conveyors are useful model elements for representing pipelines or processes that take a certain amount of time to complete. However, adding a leakage flow to a conveyor can make it difficult to initialize a model in steady-state. The following discussion will explain how to initialize conveyors with leakage in steady-state. Please refer to the model structure below while reading this discussion.

These additional variables will be also used:

*transit_time* = TRANSTIME(*conveyor*)

*conveyor_length* = *transit_time*/DT

*leakage_fraction* = the user-specified leakage fraction

**Linear Leakage**

The default leakage is linear in behavior. The total amount that leaks across the length of the conveyor is directly proportional to the inflowing amount. The leakage fraction is the constant of proportionality. Thus, the *fraction* of inflowing material that makes it to the conveyor’s outflow is exactly

1 – *leakage_fraction*

Given the sample model structure above, to achieve equilibrium, *conveyor_outflow* must equal *outflow*. For this to happen, we need to set the *inflow* as follows:

*inflow* = *outflow*/(1 – *leakage_fraction*)

The conveyor’s steady-state value is then:

*conveyor* = *transit_time***inflow* – (*conveyor_length* – 1)**leakage**DT/2

where the initial value of *leakage* is:

*leakage* = *leakage_fraction***inflow*

This must be calculated outside the program and entered as a constant into the conveyor as conveyors cannot be given equations (they can, however, be set to a the value of a single converter, but you must be careful how you calculate this to avoid circularity).

**Exponential Leakage**

Optionally, leakage can be made exponential. The amount that leaks each DT is proportional to the amount remaining in the conveyor. In this case, the leakage fraction is the fraction that leaks each unit of time so, for long conveyors, a lot of material can leak away. Given the transit time, the *fraction* of inflowing material that makes it to the conveyor’s outflow is approximately

1 – (1 – *leakage_fraction*)^*transit_time*

Given the sample model structure above, to achieve equilibrium, *conveyor_outflow* must equal *outflow*. For this to happen, we need to set the *inflow* as follows:

*per_dt_no_leak* = 1 – *leakage_fraction**DT

*inflow* = *outflow*/(*per_dt_no_leak*^*conveyor_length*)

For steady-state, the conveyor itself must then be set as follows:

*conveyor* = (*inflow**DT)*(1 – *per_dt_no_leak*^*conveyor_length*)/(1 – *per_dt_no_leak*)

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