Pipe (MA)

Rigid conduit for moist air flow

• Library:
• Simscape / Foundation Library / Moist Air / Elements

Description

The Pipe (MA) block models pipe flow dynamics in a moist air network due to viscous friction losses and convective heat transfer with the pipe wall. The pipe contains a constant volume of moist air. The pressure and temperature evolve based on the compressibility and thermal capacity of this moist air volume. Liquid water condenses out of the moist air volume when it reaches saturation. Choked flow occurs when the outlet reaches sonic condition.

Caution

Air flow through this block can choke. If a Mass Flow Rate Source (MA) block or a Controlled Mass Flow Rate Source (MA) block connected to the Pipe (MA) block specifies a greater mass flow rate than the possible choked mass flow rate, the simulation generates an error. For more information, see Choked Flow.

The block equations use these symbols. Subscripts `a`, `w`, and `g` indicate the properties of dry air, water vapor, and trace gas, respectively. Subscript `ws` indicates water vapor at saturation. Subscripts `A`, `B`, `H`, and `S` indicate the appropriate port. Subscript `I` indicates the properties of the internal moist air volume.

 $\stackrel{˙}{m}$ Mass flow rate Φ Energy flow rate Q Heat flow rate p Pressure ρ Density R Specific gas constant V Volume of moist air inside the pipe cv Specific heat at constant volume cp Specific heat at constant pressure h Specific enthalpy u Specific internal energy x Mass fraction (xw is specific humidity, which is another term for water vapor mass fraction) y Mole fraction φ Relative humidity r Humidity ratio T Temperature t Time

Mass and Energy Balance

The net flow rates into the moist air volume inside the pipe are

`$\begin{array}{l}{\stackrel{˙}{m}}_{net}={\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}-{\stackrel{˙}{m}}_{condense}+{\stackrel{˙}{m}}_{wS}+{\stackrel{˙}{m}}_{gS}\\ {\Phi }_{net}={\Phi }_{A}+{\Phi }_{B}+{Q}_{H}-{\Phi }_{condense}+{\Phi }_{S}\\ {\stackrel{˙}{m}}_{w,net}={\stackrel{˙}{m}}_{wA}+{\stackrel{˙}{m}}_{wB}-{\stackrel{˙}{m}}_{condense}+{\stackrel{˙}{m}}_{wS}\\ {\stackrel{˙}{m}}_{g,net}={\stackrel{˙}{m}}_{gA}+{\stackrel{˙}{m}}_{gB}+{\stackrel{˙}{m}}_{gS}\end{array}$`

where:

• $\stackrel{˙}{m}$condense is the rate of condensation.

• Φcondense is the rate of energy loss from the condensed water.

• ΦS is the rate of energy added by the sources of moisture and trace gas. ${\stackrel{˙}{m}}_{wS}$ and ${\stackrel{˙}{m}}_{gS}$ are the mass flow rates of water and gas, respectively, through port S. The values of ${\stackrel{˙}{m}}_{wS}$, ${\stackrel{˙}{m}}_{gS}$, and ΦS are determined by the moisture and trace gas sources connected to port S of the pipe.

Water vapor mass conservation relates the water vapor mass flow rate to the dynamics of the moisture level in the internal moist air volume:

`$\frac{d{x}_{wI}}{dt}{\rho }_{I}V+{x}_{wI}{\stackrel{˙}{m}}_{net}={\stackrel{˙}{m}}_{w,net}$`

Similarly, trace gas mass conservation relates the trace gas mass flow rate to the dynamics of the trace gas level in the internal moist air volume:

`$\frac{d{x}_{gI}}{dt}{\rho }_{I}V+{x}_{gI}{\stackrel{˙}{m}}_{net}={\stackrel{˙}{m}}_{g,net}$`

Mixture mass conservation relates the mixture mass flow rate to the dynamics of the pressure, temperature, and mass fractions of the internal moist air volume:

`$\left(\frac{1}{{p}_{I}}\frac{d{p}_{I}}{dt}-\frac{1}{{T}_{I}}\frac{d{T}_{I}}{dt}\right){\rho }_{I}V+\frac{{R}_{a}-{R}_{w}}{{R}_{I}}\left({\stackrel{˙}{m}}_{w,net}-{x}_{w}{\stackrel{˙}{m}}_{net}\right)+\frac{{R}_{a}-{R}_{g}}{{R}_{I}}\left({\stackrel{˙}{m}}_{g,net}-{x}_{g}{\stackrel{˙}{m}}_{net}\right)={\stackrel{˙}{m}}_{net}$`

Finally, energy conservation relates the energy flow rate to the dynamics of the pressure, temperature, and mass fractions of the internal moist air volume:

`${\rho }_{I}{c}_{vI}V\frac{d{T}_{I}}{dt}+\left({u}_{wI}-{u}_{aI}\right)\left({\stackrel{˙}{m}}_{w,net}-{x}_{w}{\stackrel{˙}{m}}_{net}\right)+\left({u}_{gI}-{u}_{aI}\right)\left({\stackrel{˙}{m}}_{g,net}-{x}_{g}{\stackrel{˙}{m}}_{net}\right)+{u}_{I}{\stackrel{˙}{m}}_{net}={\Phi }_{net}$`

The equation of state relates the mixture density to the pressure and temperature:

`${p}_{I}={\rho }_{I}{R}_{I}{T}_{I}$`

The mixture specific gas constant is

`${R}_{I}={x}_{aI}{R}_{a}+{x}_{wI}{R}_{w}+{x}_{gI}{R}_{g}$`

Momentum Balance

The momentum balance for each half of the pipe models the pressure drop due to momentum flux and viscous friction:

`$\begin{array}{l}{p}_{A}-{p}_{I}={\left(\frac{{\stackrel{˙}{m}}_{A}}{S}\right)}^{2}\cdot \left(\frac{{T}_{I}}{{p}_{I}}-\frac{{T}_{A}}{{p}_{A}}\right){R}_{I}+\Delta {p}_{AI}\\ {p}_{B}-{p}_{I}={\left(\frac{{\stackrel{˙}{m}}_{B}}{S}\right)}^{2}\cdot \left(\frac{{T}_{I}}{{p}_{I}}-\frac{{T}_{B}}{{p}_{B}}\right){R}_{I}+\Delta {p}_{BI}\end{array}$`

where:

• p is the pressure at port A, port B, or internal node I, as indicated by the subscript.

• ρ is the density at port A, port B, or internal node I, as indicated by the subscript.

• S is the cross-sectional area of the pipe.

• ΔpAI and ΔpBI are pressure losses due to viscous friction.

The pressure losses due to viscous friction, ΔpAI and ΔpBI, depend on the flow regime. The Reynolds numbers for each half of the pipe are defined as:

`$\begin{array}{l}{\mathrm{Re}}_{A}=\frac{|{\stackrel{˙}{m}}_{A}|\cdot {D}_{h}}{S\cdot {\mu }_{I}}\\ {\mathrm{Re}}_{B}=\frac{|{\stackrel{˙}{m}}_{B}|\cdot {D}_{h}}{S\cdot {\mu }_{I}}\end{array}$`

where:

• Dh is the hydraulic diameter of the pipe.

• μI is the dynamic viscosity at the internal node.

If the Reynolds number is less than the value of the Laminar flow upper Reynolds number limit parameter, then the flow is in the laminar flow regime. If the Reynolds number is greater than the value of the Turbulent flow lower Reynolds number limit parameter, then the flow is in the turbulent flow regime.

In the laminar flow regime, the pressure losses due to viscous friction are:

`$\begin{array}{l}\Delta {p}_{A{I}_{lam}}={f}_{shape}\frac{{\stackrel{˙}{m}}_{A}\cdot {\mu }_{I}}{2{\rho }_{I}\cdot {D}_{h}^{2}\cdot S}\cdot \frac{L+{L}_{eqv}}{2}\\ \Delta {p}_{B{I}_{lam}}={f}_{shape}\frac{{\stackrel{˙}{m}}_{B}\cdot {\mu }_{I}}{2{\rho }_{I}\cdot {D}_{h}^{2}\cdot S}\cdot \frac{L+{L}_{eqv}}{2}\end{array}$`

where:

• fshape is the value of the Shape factor for laminar flow viscous friction parameter.

• Leqv is the value of the Aggregate equivalent length of local resistances parameter.

In the turbulent flow regime, the pressure losses due to viscous friction are:

`$\begin{array}{l}\Delta {p}_{A{I}_{tur}}={f}_{Darc{y}_{A}}\frac{{\stackrel{˙}{m}}_{A}\cdot |{\stackrel{˙}{m}}_{A}|}{2{\rho }_{I}\cdot {D}_{h}\cdot {S}^{2}}\cdot \frac{L+{L}_{eqv}}{2}\\ \Delta {p}_{B{I}_{tur}}={f}_{Darc{y}_{B}}\frac{{\stackrel{˙}{m}}_{B}\cdot |{\stackrel{˙}{m}}_{B}|}{2{\rho }_{I}\cdot {D}_{h}\cdot {S}^{2}}\cdot \frac{L+{L}_{eqv}}{2}\end{array}$`

where fDarcy is the Darcy friction factor at port A or B, as indicated by the subscript.

The Darcy friction factors are computed from the Haaland correlation:

`$\begin{array}{l}{f}_{Darc{y}_{A}}={\left[-1.8\mathrm{log}\left(\frac{6.9}{{\mathrm{Re}}_{A}}+{\left(\frac{{\epsilon }_{rough}}{3.7{D}_{h}}\right)}^{1.11}\right)\right]}^{-2}\\ {f}_{Darc{y}_{B}}={\left[-1.8\mathrm{log}\left(\frac{6.9}{{\mathrm{Re}}_{B}}+{\left(\frac{{\epsilon }_{rough}}{3.7{D}_{h}}\right)}^{1.11}\right)\right]}^{-2}\end{array}$`

where εrough is the value of the Internal surface absolute roughness parameter.

When the Reynolds number is between the Laminar flow upper Reynolds number limit and the Turbulent flow lower Reynolds number limit parameter values, the flow is in transition between laminar flow and turbulent flow. The pressure losses due to viscous friction during the transition region follow a smooth connection between those in the laminar flow regime and those in the turbulent flow regime.

The heat exchanged with the pipe wall through port H is added to the energy of the moist air volume represented by the internal node via the energy conservation equation (see Mass and Energy Balance). Therefore, the momentum balances for each half of the pipe, between port A and the internal node and between port B and the internal node, are assumed to be adiabatic processes. The adiabatic relations are:

`$\begin{array}{l}{h}_{A}-{h}_{I}={\left(\frac{{R}_{I}{\stackrel{˙}{m}}_{A}}{S}\right)}^{2}\left[{\left(\frac{{T}_{I}}{{p}_{I}}\right)}^{2}-{\left(\frac{{T}_{A}}{{p}_{A}}\right)}^{2}\right]\\ {h}_{B}-{h}_{I}={\left(\frac{{R}_{I}{\stackrel{˙}{m}}_{B}}{S}\right)}^{2}\left[{\left(\frac{{T}_{I}}{{p}_{I}}\right)}^{2}-{\left(\frac{{T}_{B}}{{p}_{B}}\right)}^{2}\right]\end{array}$`

where h is the specific enthalpy at port A, port B, or internal node I, as indicated by the subscript.

Convective Heat Transfer

The convective heat transfer equation between the pipe wall and the internal moist air volume is:

`${Q}_{H}={Q}_{conv}+\frac{{k}_{I}{S}_{surf}}{{D}_{h}}\left({T}_{H}-{T}_{I}\right)$`

Ssurf is the pipe surface area, Ssurf = 4SL/Dh. Assuming an exponential temperature distribution along the pipe, the convective heat transfer is

`${Q}_{conv}=|{\stackrel{˙}{m}}_{avg}|{c}_{{p}_{avg}}\left({T}_{H}-{T}_{in}\right)\left(1-\mathrm{exp}\left(-\frac{{h}_{coeff}{S}_{surf}}{|{\stackrel{˙}{m}}_{avg}|{c}_{{p}_{avg}}}\right)\right)$`

where:

• Tin is the inlet temperature depending on flow direction.

• ${\stackrel{˙}{m}}_{avg}=\left({\stackrel{˙}{m}}_{A}-{\stackrel{˙}{m}}_{B}\right)/2$ is the average mass flow rate from port A to port B.

• ${c}_{{p}_{avg}}$ is the specific heat evaluated at the average temperature.

The heat transfer coefficient, hcoeff, depends on the Nusselt number:

`${h}_{coeff}=Nu\frac{{k}_{avg}}{{D}_{h}}$`

where kavg is the thermal conductivity evaluated at the average temperature. The Nusselt number depends on the flow regime. The Nusselt number in the laminar flow regime is constant and equal to the value of the Nusselt number for laminar flow heat transfer parameter. The Nusselt number in the turbulent flow regime is computed from the Gnielinski correlation:

`$N{u}_{tur}=\frac{\frac{{f}_{Darcy}}{8}\left({\mathrm{Re}}_{avg}-1000\right){\mathrm{Pr}}_{avg}}{1+12.7\sqrt{\frac{{f}_{Darcy}}{8}}\left({\mathrm{Pr}}_{avg}^{2/3}-1\right)}$`

where Pravg is the Prandtl number evaluated at the average temperature. The average Reynolds number is

`${\mathrm{Re}}_{avg}=\frac{|{\stackrel{˙}{m}}_{avg}|{D}_{h}}{S{\mu }_{avg}}$`

where μavg is the dynamic viscosity evaluated at the average temperature. When the average Reynolds number is between the Laminar flow upper Reynolds number limit and the Turbulent flow lower Reynolds number limit parameter values, the Nusselt number follows a smooth transition between the laminar and turbulent Nusselt number values.

Saturation and Condensation

When the moist air volume reaches saturation, condensation may occur. The specific humidity at saturation is

`${x}_{wsI}={\phi }_{ws}\frac{{R}_{I}}{{R}_{w}}\frac{{p}_{wsI}}{{p}_{I}}$`

where:

• φws is the relative humidity at saturation (typically 1).

• pwsI is the water vapor saturation pressure evaluated at TI.

The rate of condensation is

where τcondense is the value of the Condensation time constant parameter.

The condensed water is subtracted from the moist air volume, as shown in the conservation equations. The energy associated with the condensed water is

`${\Phi }_{condense}={\stackrel{˙}{m}}_{condense}\left({h}_{wI}-\Delta {h}_{vapI}\right)$`

where ΔhvapI is the specific enthalpy of vaporization evaluated at TI.

Other moisture and trace gas quantities are related to each other as follows:

`$\begin{array}{l}{\phi }_{wI}=\frac{{y}_{wI}{p}_{I}}{{p}_{wsI}}\\ {y}_{wI}=\frac{{x}_{wI}{R}_{w}}{{R}_{I}}\\ {r}_{wI}=\frac{{x}_{wI}}{1-{x}_{wI}}\\ {y}_{gI}=\frac{{x}_{gI}{R}_{g}}{{R}_{I}}\\ {x}_{aI}+{x}_{wI}+{x}_{gI}=1\end{array}$`

Choked Flow

The unchoked pressure at port A or B is the value of the corresponding Across variable at that port:

`$\begin{array}{l}{p}_{{A}_{unchoked}}=\text{A}\text{.p}\\ {p}_{{B}_{unchoked}}=\text{B}\text{.p}\end{array}$`

However, the port pressure variables used in the momentum balance equations, pA and pB, do not necessarily coincide with the pressure across variables `A.p` and `B.p` because the pipe outlet may choke. Choked flow occurs when the downstream pressure is sufficiently low. At that point, the flow depends only on the conditions at the inlet. Therefore, when choked, the outlet pressure (pA or pB, whichever is the outlet) cannot decrease further even if the pressure downstream, represented by `A.p` or `B.p`, continues to decrease.

Choking can occur at the pipe outlet, but not at the pipe inlet. Therefore, if port A is the inlet, then pA = `A.p`. If port A is the outlet, then

Similarly, if port B is the inlet, then pB = `B.p`. If port B is the outlet, then

The choked pressures at ports A and B are derived from the momentum balance by assuming the outlet velocity is equal to the speed of sound:

`$\begin{array}{l}{p}_{{A}_{choked}}-{p}_{I}={p}_{{A}_{choked}}\left(\frac{{p}_{{A}_{choked}}{T}_{I}}{{p}_{I}{T}_{A}}-1\right)\frac{{c}_{{p}_{A}}}{{c}_{{v}_{I}}}+\Delta {p}_{AI}\\ {p}_{{B}_{choked}}-{p}_{I}={p}_{{B}_{choked}}\left(\frac{{p}_{{B}_{choked}}{T}_{I}}{{p}_{I}{T}_{B}}-1\right)\frac{{c}_{{p}_{B}}}{{c}_{{v}_{I}}}+\Delta {p}_{BI}\end{array}$`

Variables

To set the priority and initial target values for the block variables prior to simulation, use the Variables tab in the block dialog box (or the Variables section in the block Property Inspector). For more information, see Set Priority and Initial Target for Block Variables and Initial Conditions for Blocks with Finite Moist Air Volume.

Assumptions and Limitations

• The pipe wall is perfectly rigid.

• The flow is fully developed. Friction losses and heat transfer do not include entrance effects.

• The effect of gravity is negligible.

• Fluid inertia is negligible.

• This block does not model supersonic flow.

Ports

Output

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Physical signal output port that measures the rate of condensation in the pipe.

Physical signal output port that outputs a vector signal. The vector contains the pressure (in Pa), temperature (in K), moisture level, and trace gas level measurements inside the component. Use the Measurement Selector (MA) block to unpack this vector signal.

Conserving

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Moist air conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.

Moist air conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.

Thermal conserving port associated with the temperature of the pipe wall. The block includes the convective heat transfer between the moist air mixture inside the pipe and the pipe wall.

Connect this port to port S of a block from the Moisture & Trace Gas Sources library to add or remove moisture and trace gas. For more information, see Using Moisture and Trace Gas Sources.

Dependencies

This port is visible only if you set the Moisture and trace gas source parameter to `Controlled`.

Parameters

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Main

Length of the pipe along the direction of flow.

Internal area of the pipe normal to the direction of the flow.

Diameter of an equivalent cylindrical pipe with the same cross-sectional area.

Friction and Heat Transfer

Combined length of all local resistances present in the pipe. Local resistances include bends, fittings, armatures, and pipe inlets and outlets. The effect of the local resistances is to increase the effective length of the pipe segment. This length is added to the geometrical pipe length only for friction calculations. The moist air volume depends only on the pipe geometrical length, defined by the Pipe length parameter.

Average depth of all surface defects on the internal surface of the pipe, which affects the pressure loss in the turbulent flow regime.

Reynolds number above which flow begins to transition from laminar to turbulent. This number equals the maximum Reynolds number corresponding to the fully developed laminar flow.

Reynolds number below which flow begins to transition from turbulent to laminar. This number equals to the minimum Reynolds number corresponding to the fully developed turbulent flow.

Dimensionless factor that encodes the effect of pipe cross-sectional geometry on the viscous friction losses in the laminar flow regime. Typical values are 64 for a circular cross section, 57 for a square cross section, 62 for a rectangular cross section with an aspect ratio of 2, and 96 for a thin annular cross section [1].

Ratio of convective to conductive heat transfer in the laminar flow regime. Its value depends on the pipe cross-sectional geometry and pipe wall thermal boundary conditions, such as constant temperature or constant heat flux. Typical value is 3.66, for a circular cross section with constant wall temperature [2].

Moisture and Trace Gas

Relative humidity above which condensation occurs.

Characteristic time scale at which an oversaturated moist air volume returns to saturation by condensing out excess moisture.

This parameter controls visibility of port S and provides these options for modeling moisture and trace gas levels inside the component:

• `None` — No moisture or trace gas is injected into or extracted from the block. Port S is hidden. This is the default.

• `Constant` — Moisture and trace gas are injected into or extracted from the block at a constant rate. The same parameters as in the Moisture Source (MA) and Trace Gas Source (MA) blocks become available in the Moisture and Trace Gas section of the block interface. Port S is hidden.

• `Controlled` — Moisture and trace gas are injected into or extracted from the block at a time-varying rate. Port S is exposed. Connect the Controlled Moisture Source (MA) and Controlled Trace Gas Source (MA) blocks to this port.

Select whether the block adds or removes moisture as water vapor or liquid water:

• `Vapor` — The enthalpy of the added or removed moisture corresponds to the enthalpy of water vapor, which is greater than that of liquid water.

• `Liquid` — The enthalpy of the added or removed moisture corresponds to the enthalpy of liquid water, which is less than that of water vapor.

Dependencies

Enabled when the Moisture and trace gas source parameter is set to `Constant`.

Water vapor mass flow rate through the block. A positive value adds moisture to the connected moist air volume. A negative value extracts moisture from that volume.

Dependencies

Enabled when the Moisture and trace gas source parameter is set to `Constant`.

Select a specification method for the moisture temperature:

• `Atmospheric temperature` — Use the atmospheric temperature, specified by the Moist Air Properties (MA) block connected to the circuit.

• `Specified temperature` — Specify a value by using the Temperature of added moisture parameter.

Dependencies

Enabled when the Moisture and trace gas source parameter is set to `Constant`.

Enter the desired temperature of added moisture. This temperature remains constant during simulation. The block uses this value to evaluate the specific enthalpy of the added moisture only. The specific enthalpy of removed moisture is based on the temperature of the connected moist air volume.

Dependencies

Enabled when the Added moisture temperature specification parameter is set to `Specified temperature`.

Trace gas mass flow rate through the block. A positive value adds trace gas to the connected moist air volume. A negative value extracts trace gas from that volume.

Dependencies

Enabled when the Moisture and trace gas source parameter is set to `Constant`.

Select a specification method for the trace gas temperature:

• `Atmospheric temperature` — Use the atmospheric temperature, specified by the Moist Air Properties (MA) block connected to the circuit.

• `Specified temperature` — Specify a value by using the Temperature of added trace gas parameter.

Dependencies

Enabled when the Moisture and trace gas source parameter is set to `Constant`.

Enter the desired temperature of added trace gas. This temperature remains constant during simulation. The block uses this value to evaluate the specific enthalpy of the added trace gas only. The specific enthalpy of removed trace gas is based on the temperature of the connected moist air volume.

Dependencies

Enabled when the Added trace gas temperature specification parameter is set to `Specified temperature`.

References

[1] White, F. M., Fluid Mechanics. 7th Ed, Section 6.8. McGraw-Hill, 2011.

[2] Cengel, Y. A., Heat and Mass Transfer – A Practical Approach. 3rd Ed, Section 8.5. McGraw-Hill, 2007.