Turbine for boosted engines
Powertrain Blockset / Propulsion / Combustion Engine Components / Boost
The Turbine block uses the conservation of mass and energy to calculate mass and heat flow rates for turbines with either fixed or variable geometry. You can configure the block with a wastegate valve to bypass the turbine. The block uses twoway ports to connect to the inlet and outlet control volumes and the drive shaft. You can specify the lookup tables to calculate the mass flow rate and turbine efficiency. Typically, turbine manufacturers provide the mass flow rate and efficiency tables as a function of corrected speed and pressure ratio. The block does not support reverse mass flow.
If you have ModelBased Calibration Toolbox™, click Calibrate Performance Maps to virtually calibrate the mass flow rate and turbine efficiency lookup tables using measured data.
The mass flows from the inlet control volume to outlet control volume.
The Turbine block implements equations to model the performance, wastegate flow, and combined flow.
If you have ModelBased Calibration Toolbox, click Calibrate Performance Maps to virtually calibrate the corrected mass flow rate and turbine efficiency lookup tables using measured data. The dialog box steps through these tasks.
Task  Description  

Import turbine data  Import this turbine data from a file. For more information, see Using Data (ModelBased Calibration Toolbox).
ModelBased Calibration Toolbox limits the speed and pressure ratio breakpoint values to the maximum values in the file. To filter or edit the data, select Edit in Application. The ModelBased Calibration Toolbox Data Editor opens.  
Generate response models  ModelBased Calibration Toolbox fits the imported data and generates response models.
To assess or adjust the response model fit, select Edit in Application. The ModelBased Calibration Toolbox Model Browser opens. For more information, see Model Assessment (ModelBased Calibration Toolbox).  
Generate calibration  ModelBased Calibration Toolbox calibrates the response model and generates calibrated tables.
To assess or adjust the calibration, select Edit in Application. The ModelBased Calibration Toolbox CAGE Browser opens. For more information, see Calibration Lookup Tables (ModelBased Calibration Toolbox).  
Update block parameters  Update these corrected mass flow rate and efficiency parameters with the calibration.

The block uses these equations to model the thermodynamics.
Calculation  Equations 

Forward mass flow  ${\dot{m}}_{turb}>0$ ${p}_{01}={p}_{inlet}$ ${p}_{02}={p}_{outlet}$ ${T}_{01}={T}_{inlet}$ ${h}_{01}={h}_{inlet}$ 
First law of thermodynamics  ${\dot{W}}_{turb}={\dot{m}}_{turb}{c}_{p}\left({T}_{01}{T}_{02}\right)$ 
Isentropic efficiency  ${\eta}_{turb}=\frac{{h}_{01}{h}_{02}}{{h}_{01}{h}_{02\text{s}}}=\frac{{T}_{01}{T}_{02}}{{T}_{01}{T}_{02\text{s}}}$ 
Isentropic outlet temperature, assuming ideal gas, and constant specific heats  ${T}_{02s}={T}_{01}{\left(\frac{{p}_{02}}{{p}_{01}}\right)}^{\frac{\gamma 1}{\gamma}}$ 
Specific heat ratio  $\gamma =\frac{{c}_{p}}{{c}_{p}R}$ 
Outlet temperature  ${T}_{02}={T}_{01}+{\eta}_{turb}{T}_{01}\left\{1{\left(\frac{{p}_{02}}{{p}_{01}}\right)}^{\frac{\gamma 1}{\gamma}}\right\}$ 
Heat flows  ${q}_{in,turb}={\dot{m}}_{turb}{c}_{p}{T}_{01}$ ${q}_{out,turb}={\dot{m}}_{turb}{c}_{p}{T}_{02}$ 
Drive shaft torque  ${\tau}_{turb}=\frac{{\dot{W}}_{turb}}{\omega}$ 
The equations use these variables.
${p}_{\text{inlet}}$,${p}_{01}$  Inlet control volume total pressure 
${T}_{inlet}$, ${T}_{01}$  Inlet control volume total temperature 
${h}_{inlet}$, ${h}_{01}$  Inlet control volume total specific enthalpy 
${p}_{outlet}$, ${p}_{02}$  Outlet control volume total pressure 
${T}_{outlet}$  Outlet control volume total temperature 
${h}_{outlet}$  Outlet control volume total specific enthalpy 
${\dot{W}}_{turb}$  Drive shaft power 
${T}_{02}$  Temperature exiting the turbine 
${h}_{02}$  Outlet total specific enthalpy 
${\dot{m}}_{turb}$  Turbine mass flow rate 
${q}_{in,turb}$  Turbine inlet heat flow rate 
${q}_{out,turb}$  Turbine outlet heat flow rate 
${\eta}_{turb}$  Turbine isentropic efficiency 
${T}_{02s}$  Isentropic outlet total temperature 
${h}_{02s}$  Isentropic outlet total specific enthalpy 
$R$  Ideal gas constant 
${c}_{p}$  Specific heat at constant pressure 
$\gamma $  Specific heat ratio 
${\tau}_{turb}$  Drive shaft torque 
The block implements lookup tables based on these equations.
Calculation  Equation  

Corrected mass flow rate  ${\dot{m}}_{corr}={\dot{m}}_{turb}\frac{\sqrt{{T}_{01}/{T}_{ref}}}{{p}_{01}/{p}_{ref}}$  
Corrected speed  ${\omega}_{corr}=\frac{\omega}{\sqrt{{T}_{01}/{T}_{ref}}}$  
Pressure expansion ratio  ${p}_{r}=\frac{{p}_{01}}{{p}_{02}}$  
Efficiency lookup table  Fixed geometry (3D table)  ${\eta}_{turbfx,tbl}=f\left({\omega}_{corr},{p}_{r}\right)$ 
Variable geometry (3D table)  ${\eta}_{turbvr,tbl}=f\left({\omega}_{corr},{p}_{r},{L}_{rack}\right)$  
Corrected mass flow lookup table  Fixed geometry (3D table)  ${\dot{m}}_{corrfx,tbl}=f\left({\omega}_{corr},{p}_{r}\right)$ 
Variable geometry (3D table)  ${\dot{m}}_{corrvr,tbl}=f\left({\omega}_{corr},{p}_{r},{L}_{rack}\right)$ 
The equations use these variables.
${p}_{01}$  Inlet control volume total pressure 
${p}_{r}$  Pressure expansion ratio 
${p}_{02}$  Outlet control volume total pressure 
${P}_{ref}$  Lookup table reference pressure 
${T}_{01}$  Inlet control volume total temperature 
${T}_{ref}$  Lookup table reference temperature 
${\dot{m}}_{turb}$  Turbine mass flow rate 
$\omega $  Drive shaft speed 
${\omega}_{corr}$  Corrected drive shaft speed 
${L}_{rack}$  Variable geometry turbine rack position 
${\eta}_{turbfx,tbl}$  Efficiency 3D lookup table for fixed geometry 
${\dot{m}}_{corrfx,tbl}$  Corrected mass flow rate 3D lookup table for fixed geometry 
${\eta}_{turbvr,tbl}$  Efficiency 3D lookup table for variable geometry 
${\dot{m}}_{corrvr,tbl}$  Corrected mass flow rate 3D lookup table for variable geometry 
To calculate the wastegate heat and mass flow rates, the Turbine block uses a Flow Restriction block. The Flow Restriction block uses the wastegate flow area.
${A}_{wg}={A}_{wgpctcmd}\frac{{A}_{wgopen}}{100}$
The equation uses these variables.
${A}_{wgpctcmd}$  Wastegate valve area percent command 
${A}_{wg}$  Wastegate valve area 
${A}_{wgopen}$  Wastegate valve area when fully open 
To represent flow through the wastegate valve and turbine, the block uses these equations.
Calculation  Equations  

Blocks not configured with a wastegate valve  ${\dot{m}}_{wg}={q}_{wg}=0$  
Total mass flow rate  ${\dot{m}}_{total}={\dot{m}}_{turb}+{\dot{m}}_{wg}$  
Total heat flow rate  ${q}_{inlet}={q}_{in,turb}+{q}_{wg}$ ${q}_{outlet}={q}_{out,turb}+{q}_{wg}$  
Combined temperature exiting the wastegate valve and turbine  ${T}_{outflw}=\{\begin{array}{cc}\frac{{q}_{outlet}}{{\dot{m}}_{total}{c}_{p}}& {\dot{m}}_{total}>{\dot{m}}_{thresh}\\ \frac{{T}_{02}+{T}_{outflw,wg}}{2}& else\end{array}$ 
The block uses the internal signal FlwDir
to track the direction of the flow.
The equations use these variables.
${\dot{m}}_{total}$  Total mass flow rate through the wastegate valve and turbine 
${\dot{m}}_{turb}$  Turbine mass flow rate 
${\dot{m}}_{wg}$  Mass flow rate through the wastegate valve 
${q}_{inlet}$  Total inlet heat flow rate 
${q}_{outlet}$  Total outlet heat flow rate 
${q}_{in,turb}$  Turbine inlet heat flow rate 
${q}_{out,turb}$  Turbine outlet heat flow rate 
$${q}_{wg}$$  Wastegate valve heat flow rate 
${T}_{02}$  Temperature exiting the turbine 
${T}_{outflw}$  Total temperature exiting the block 
${T}_{outflw,wg}$  Temperature exiting the wastegate valve 
${\dot{m}}_{thresh}$  Mass flow rate threshold to prevent dividing by zero 
${c}_{p}$  Specific heat at constant pressure 
For the power accounting, the block implements these equations.
Bus Signal  Description  Equations  


 PwrDriveshft  Power transmitted from the shaft  ${\dot{W}}_{turb}$ 
 Heat flow rate at port A  ${q}_{outlet}$  
PwrHeatFlwOut  Heat flow rate at port B  ${q}_{outlet}$  
 PwrLoss  Power loss  ${q}_{inlet}{q}_{outlet}+{\dot{W}}_{turb}$  
 Not used 
The equations use these variables.
${\dot{W}}_{turb}$  Drive shaft power 
${q}_{outlet}$  Total outlet heat flow rate 
${q}_{inlet}$  Total inlet heat flow rate 
[1] Heywood, John B. Internal Combustion Engine Fundamentals. New York: McGrawHill, 1988.
[2] Eriksson, Lars and Lars Nielsen. Modeling and Control of Engines and Drivelines. Chichester, West Sussex, United Kingdom: John Wiley & Sons Ltd, 2014.