Question

4. Write a Matlab code that computes the Reynolds number, discharge coefficient, in the discharge coefficient, and air flow rate for a given orifice plate geometry an difference. Your code should be able to compute the flow rate (mass and ΔΡ across an orifice plate. You will continue to use your code for the rest example set of data against which you can test your code is provided in from your code must be in the same units as shown in Table 1. At a mini compute the following using the information from the left column of Tabl ed semester. An Table 1. The model output volumetric or a measur of the . Reynolds number Rep, e discharge coefficient C e relative uncertainty in the discharge coefficient 6C/C e volumetric flow rate V in m3/s and CFM mass flow rate m in kg/m3, and be The left column of Table 1 are your model inputs and model constants. The model inputs can changed by your GTA when evaluating the code. The model constants can be hard-coded and do not need to change. You must include comments in your Matlab code that define each variabl nd the units on that variable. Your GTAs will inspect and test your code. This exercise is pass fail. You receive no credit unless your code passes inspection and testing. The criteria successful code passing i for (a) Inspection: By quick inspection, your GTA can easily interp ret your variable names to know which ones correspond to the minimum list of parameters described above. For example, if your GTA has to guess which variable is mass flow rate because there is no comment defining the variable, then your code fails. You must comment your code appropriately (b) Testing: You will upload your Matlab code as a *.m file, which your GTA will test using a second test case different from that shown in Table 1. If your Matlab code runs without error and correctly outputs the minimum list of parameters using a second test case (not shown here), then the code passes. If your code can successfully output the right column of Table 1, then there should be no problem in passing the GTA test The challenge with this exercise, unlike last week, is that you do not have a Reynolds number with which to compute the discharge coefficient C. This week you have too many unknowns and cannot compute C directly. Flow rate is required to calculate the Reynolds number; Reynolds number is required to calculate the discharge coefficient C: C is required to calculate the flow rate. This circular solution is a common occurrence in engineering. What you must do is make an intelligent initial guess on one of the unknowns (in this case the Reynolds number), then run through the computations. Recompute a new Reynolds number based on the results and repeat until your answer no longer changes. Figure 1 illustrates the recursive solving logic for computing flow rate.

Answer 1

for reynolds number

function [ Re ] = Re_func(V_inf, altitude, l)
%This function outputs the Reynolds number for dry air given an altitude [m],
%free stream velocity [ m/s], and characteristic length [ m]. Often the characteristic
%length for airfoils is the chord length.
%Inputs = altitude [m], V_inf = free stream velocity [m/s], l = characteristic length [m]
%Outputs = Reynolds numbers [ dimensionless ]
% arg2 and arg3 are optional,

Orifices

Description

The block represents a variable orifice of any type as a data-sheet-based model. Depending on data listed in the manufacturer's catalogs or data sheets for your particular orifice, you can choose one of the following model parameterization options:

• By maximum area and opening — Use this option if the data sheet provides only the orifice maximum area and the control member maximum stroke.

• By area vs. opening table — Use this option if the catalog or data sheet provides a table of the orifice passage area based on the control member displacement A=A(h).

• By pressure-flow characteristic — Use this option if the catalog or data sheet provides a two-dimensional table of the pressure-flow characteristics q=q(p,h).

In the first case, the passage area is assumed to be linearly dependent on the control member displacement, that is, the orifice is assumed to be closed at the initial position of the control member (zero displacement), and the maximum opening takes place at the maximum displacement. In the second case, the passage area is determined by one-dimensional interpolation from the table A=A(h). In both cases, a small leakage area is assumed to exist even after the orifice is completely closed. Physically, it represents a possible clearance in the closed valve, but the main purpose of the parameter is to maintain numerical integrity of the circuit by preventing a portion of the system from getting isolated after the valve is completely closed. An isolated or “hanging” part of the system could affect computational efficiency and even cause failure of computation.

In the first and second cases, the flow rate is computed according to the following equations:

q=CD⋅A(h)G2ρΔp(Δp2+p2Cr)1/4,

Δp=pA−pB,

h=x0+x·or

For the first parameterization, the opening area (A) is a piecewise function of the control member position (h). The area saturates at its leakage value when the control member is in the fully closed position (h_min). It saturates at its maximum value when the control member is in the fully open position (h_max).

A(h)=AleakAmaxhmaxh,Amax,h≤hminh>0h≥hmax

The minimum control member position is calculated as:

hmin=hmaxAmaxAleak

For the second parameterization, the opening area is a tabulated function of control member displacement. As with the linear parameterization, the area saturates at its leakage value when the control member is in the fully closed position, and it saturates at its maximum value when the control member is in the fully open position. Between the closed and open positions:

A=f(h)

The table summarizes the parameters used in the equations.

q	Flow rate
p	Pressure differential
pA, pB	Gauge pressures at the block terminals
CD	Flow discharge coefficient
A(h)	Instantaneous orifice passage area
Amax	Orifice maximum area
hmax	Control member maximum displacement
x0	Initial opening
x	Control member displacement from initial position
h	Orifice opening
or	Orifice orientation indicator. The variable assumes +1 value if the control member displacement in the globally assigned positive direction opens the orifice, and –1 if positive motion decreases the opening.
ρ	Fluid density
Aleak	Closed orifice leakage area
pcr	Minimum pressure for turbulent flow

The minimum pressure for turbulent flow, p_cr, is calculated according to the laminar transition specification method:

• By pressure ratio — The transition from laminar to turbulent regime is defined by the following equations:

  p_cr = (p_avg + p_atm)(1 – B_lam)

  p_avg = (p_A + p_B)/2

  where

  pavg	Average pressure between the block terminals
  patm	Atmospheric pressure, 101325 Pa
  Blam	Pressure ratio at the transition between laminar and turbulent regimes (Laminar flow pressure ratio parameter value)

• By Reynolds number — The transition from laminar to turbulent regime is defined by the following equations:

  pcr=ρ2(Recr⋅νCD⋅DH)2

  DH=G4Aπ

  where

  DH	Instantaneous orifice hydraulic diameter
  ν	Fluid kinematic viscosity
  Recr	Critical Reynolds number (Critical Reynolds number parameter value)

In the third case, when an orifice is defined by its pressure-flow characteristics, the flow rate is determined by two-dimensional interpolation. In this case, neither flow regime nor leakage flow rate is taken into account, because these features are assumed to be introduced through the tabulated data. Pressure-flow characteristics are specified with three data sets: array of orifice openings, array of pressure differentials across the orifice, and matrix of flow rate values. Each value of a flow rate corresponds to a specific combination of an opening and pressure differential.

The block positive direction is from port A to port B. This means that the flow rate is positive if it flows from A to B and the pressure differential is determined as Δp=pA−pB,. Positive signal at the physical signal port S opens or closes the orifice depending on the value of the orifice orientation indicator.

Basic Assumptions and Limitations

• Fluid inertia is not taken into account.

• For orifices specified by pressure-flow characteristics (the third parameterization option), the model does not explicitly account for the flow regime or leakage flow rate, because the tabulated data is assumed to account for these characteristics.

Parameters

Model parameterization

Select one of the following methods for specifying the orifice:

• By maximum area and opening — Provide values for the maximum orifice area and the maximum orifice opening. The passage area is linearly dependent on the control member displacement, that is, the orifice is closed at the initial position of the control member (zero displacement), and the maximum opening takes place at the maximum displacement. This is the default method.

• By area vs. opening table — Provide tabulated data of orifice openings and corresponding orifice areas. The passage area is determined by one-dimensional table lookup. You have a choice of two interpolation methods and two extrapolation methods.

• By pressure-flow characteristic — Provide tabulated data of orifice openings, pressure differentials, and corresponding flow rates. The flow rate is determined by two-dimensional table lookup. You have a choice of two interpolation methods and two extrapolation methods.

Orifice maximum area

Specify the area of a fully opened orifice. The parameter value must be greater than zero. The default value is 5e-5 m^2. This parameter is used if Model parameterization is set to By maximum area and opening.

Orifice maximum opening

Specify the maximum displacement of the control member. The parameter value must be greater than zero. The default value is 5e-4 m. This parameter is used if Model parameterization is set to By maximum area and opening.

Orifice opening vector, s

Specify the vector of input values for orifice openings as a one-dimensional array. The input values vector must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for smooth interpolation. The default values, in meters, are [-0.002 0 0.002 0.005 0.015]. If Model parameterization is set to By area vs. opening table, the Tabulated orifice openings values will be used together with Tabulated orifice area values for one-dimensional table lookup. If Model parameterization is set to By pressure-flow characteristic, the Tabulated orifice openings values will be used together with Tabulated pressure differentials and Tabulated flow rates for two-dimensional table lookup.

Orifice area vector

Specify the vector of orifice areas as a one-dimensional array. The vector must be of the same size as the orifice openings vector. All the values must be positive. The default values, in m^2, are [1e-09 2.0352e-07 4.0736e-05 0.00011438 0.00034356]. This parameter is used if Model parameterization is set to By area vs. opening table.

Pressure differential vector, dp

Specify the pressure differential vector as a one-dimensional array. The vector must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for smooth interpolation. The default values, in Pa, are [-1e+07 -5e+06 -2e+06 2e+06 5e+06 1e+07]. This parameter is used if Model parameterization is set to By pressure-flow characteristic.

Volumetric flow rate table, q(s,dp)

Specify the flow rates as an m-by-n matrix, where m is the number of orifice openings and n is the number of pressure differentials. Each value in the matrix specifies flow rate taking place at a specific combination of orifice opening and pressure differential. The matrix size must match the dimensions defined by the input vectors. The default values, in m^3/s, are:

[-1e-07 -7.0711e-08 -4.4721e-08 4.4721e-08 7.0711e-08 1e-07;
 -2.0352e-05 -1.4391e-05 -9.1017e-06 9.1017e-06 1.4391e-05 2.0352e-05;
 -0.0040736 -0.0028805 -0.0018218 0.0018218 0.0028805 0.0040736;
 -0.011438 -0.0080879 -0.0051152 0.0051152 0.0080879 0.011438;
 -0.034356 -0.024293 -0.015364 0.015364 0.024293 0.034356;]

This parameter is used if Model parameterization is set to By pressure-flow characteristic.

Interpolation method

Select one of the following interpolation methods for approximating the output value when the input value is between two consecutive grid points:

• Linear — Select this option to get the best performance.

• Smooth — Select this option to produce a continuous curve (By area vs. opening table) or surface (By pressure-flow characteristic) with continuous first-order derivatives.

For more information on interpolation algorithms, see the PS Lookup Table (1D) and PS Lookup Table (2D) block reference pages.

Extrapolation method

Select one of the following extrapolation methods for determining the output value when the input value is outside the range specified in the argument list:

• Linear — Select this option to produce a curve or surface with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.

• Nearest — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.

For more information on extrapolation algorithms, see the PS Lookup Table (1D) and PS Lookup Table (2D) block reference pages.

Orifice orientation

The parameter is introduced to specify the effect of the orifice control member motion on the valve opening. The parameter can be set to one of two options: Opens in positive direction or Opens in negative direction. The value Opens in positive direction specifies an orifice whose control member opens the valve when it is shifted in the globally assigned positive direction. The parameter is extremely useful for building a multi-orifice valve with all the orifices being controlled by the same spool. The default value is Opens in positive direction.

Flow discharge coefficient

Semi-empirical parameter for orifice capacity characterization. Its value depends on the geometrical properties of the orifice, and usually is provided in textbooks or manufacturer data sheets. The default value is 0.7.

Initial opening

Orifice initial opening. The parameter can be positive (underlapped orifice), negative (overlapped orifice), or equal to zero for zero lap configuration. The value of initial opening does not depend on the orifice orientation. The default value is 0.

Leakage area

The total area of possible leaks in the completely closed orifice. The main purpose of the parameter is to maintain numerical integrity of the circuit by preventing a portion of the system from getting isolated after the valve is completely closed. The parameter value must be greater than 0. The default value is 1e-12 m^2.

Laminar transition specification

Select how the block transitions between the laminar and turbulent regimes:

• Pressure ratio — The transition from laminar to turbulent regime is smooth and depends on the value of the Laminar flow pressure ratio parameter. This method provides better simulation robustness.

• Reynolds number — The transition from laminar to turbulent regime is assumed to take place when the Reynolds number reaches the value specified by the Critical Reynolds number parameter.

Laminar flow pressure ratio

Pressure ratio at which the flow transitions between laminar and turbulent regimes. The default value is 0.999. This parameter is visible only if the Laminar transition specification parameter is set to Pressure ratio.

Critical Reynolds number

The maximum Reynolds number for laminar flow. The value of the parameter depends on the orifice geometrical profile. You can find recommendations on the parameter value in hydraulics textbooks. The default value is 12. This parameter is visible only if the Laminar transition specification parameter is set to Reynolds number.

Variables Tab

Volumetric flow rate

Flow rate specified as the fluid volume per unit time at time zero. The value entered serves as a guide during model compilation. The actual flow rate can differ if needed to satisfy all model constraints. Set the Priority level to High to prioritize the specified value during model assembly.

Pressure drop

Pressure change from port A to port B at time zero. The value entered serves as a guide during model compilation. The actual flow rate can differ if needed to satisfy all model constraints. Set the Priority level to High to prioritize the specified value during model assembly.

$arrow_right.gif$ Restricted Parameters

Global Parameters

Parameters determined by the type of working fluid:

• Fluid density

• Fluid kinematic viscosity

Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

Ports

The block has the following ports:

A

Hydraulic conserving port associated with the orifice inlet.

B

Hydraulic conserving port associated with the orifice outlet.

S

Physical signal port to control spool displacement.

The flow rate is positive if fluid flows from port A to port B. Positive signal at the physical signal port S opens or closes the orifice depending on the value of the parameter Orifice orientation.