Documentation

Pipe (TL)

Closed conduit for the transport of fluid between thermal liquid components

• Library:
• Simscape / Fluids / Thermal Liquid / Pipes & Fittings

Description

The Pipe (TL) block models the flow of a thermal liquid through a closed conduit such as a pipe. The wall of the conduit can be rigid or flexible, the latter case allowing for expansion and contraction in the radial direction, as a compliant hose or, in the life sciences, an artery might. The thermal liquid ports can be at different elevations and the vertical distance between them can be controlled (via physical signal), for example to capture the banking of an aircraft changing course.

The pressure loss across the pipe is determined as a function of friction between the fluid and the pipe and of the rise or drop in height between the ports. For enhanced precision in models with rapid flow changes (such as those associated with the water hammer effect) the block can be configured to capture the dynamic compressibility of the fluid and its inertia. Note that such effects can reduce the speed of simulation and should be used only when necessary.

The temperature change across the pipe is determined from the energy exchanges between the pipe and the remainder of the model. These exchanges include those attributed to advection and conduction of internal energy through the thermal liquid ports (A and B) and to convection of heat through the thermal port (H). The calculation captures also the differences in elevation and static pressure established during simulation between the thermal liquid ports.

If the thermal liquid is treated as compressible, the pipe can be discretized into equal segments, each containing a portion of the total fluid volume. Internal fluid volumes such as these serve a special purpose in the thermal liquid domain: they provide the computational nodes at which to compute the domain and component variables during simulation. The greater the number of pipe segments, the finer the discretization, and the more accurate the simulation results (albeit at reduced simulation speed).

Block Variants

To capture the rise in port elevation, the block provides two variants. The default option treats this quantity as a constant (specified via the Elevation gain from port A to port B block parameter). The alternative variant treats it as a variable (controlled by physical signal port El). To change block variants, right-click the block and, from its context-sensitive menu, select Simscape + Block Choices. Click the desired variant: ```Constant elevation``` or `Variable elevation`.

Parameterizations

To model the friction losses using the data best suited for a particular application, the block provides an array of friction parameterizations. Some are based on analytical expressions requiring only a small number of empirical constants; the Haaland correlation is one such expression. Others are based on tabulated data relating various quantities of interest—the Darcy friction factor against the Reynolds number, for example, or the nominal pressure drop to the nominal mass flow rate.

Analytical and tabulated parameterizations are provided also for the heat transfer between the thermal liquid and the pipe wall. Analytical parameterizations include those based on the empirical correlations of Gnielinski and Dittus-Boelter. Tabulated parameterizations include those based on data relating the Colburn factor to the Reynolds number, the Nusselt number to the Reynolds and Prandtl numbers, or the nominal temperature differential to the nominal mass flow rate.

Discretizing the Pipe

If the pipe is segmented so that it contains more than one fluid volume, then the total mass, momentum, and energy accumulation within its span are determined as sums over the volumes that the pipe contains. Segmented pipes are treated as assemblies of smaller pipes, each pipe associated with a separate instance of this block (each block configured to provide one fluid volume). The calculations described for this block apply to a pipe with a single fluid volume.

The appropriate number of pipe segments to use in a model depends partly on the time scales over which temperature and pressure disturbances tend to propagate through the pipe. Pressure waves travel the fastest (they do so at the speed of sound in the fluid) and are often the rate-limiting factor to consider. In accordance with the Nyquist sampling theorem, in order to capture an elementary sinusoidal disturbance, at least two computational nodes—and therefore pipe segments—must be available for sampling within one wavelength:

`$\frac{c}{f}=2\frac{L}{N},$`

where c is the speed of sound, f is the frequency of the disturbance (in Hertz), L is the total length of the pipe, and N is the number of pipe segments. The left-hand side represents the wavelength of the pressure disturbance and the right-hand side the length of a pipe segment—each providing one computational node to the pipe. To capture those pressure disturbances with frequencies up to a maximum fMax, the number of segments in the pipe must therefore be at least:

`$N=2L\frac{f}{c},$`

Use this expression as a loose guideline in setting the discretization of the pipe. Other modeling constraints may factor into the decision of how many pipe segments to use and even of how to model them. More pipe segments may be required, for example, to properly define a thermal boundary condition along the length of the pipe; the pipe segments are in such a case more adequately modeled explicitly, using a separate Pipe (TL) block for each (and employing its thermal port to set the thermal boundary condition).

Use Simscape data logging to access the thermal liquid properties and states at the various nodes corresponding to the pipe segments.

Mass Balance

The thermal liquid flow enters and exits the pipe via thermal liquid ports A and B. In the default case of a rigid pipe, the volume of fluid contained between these ports is fixed. If the thermal liquid is treated as incompressible, its density (at a given operating condition) is fixed also and its mass within the pipe cannot vary with time. The mass balance between the ports is, in this simple case:

`${\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}=0,$`

where $\stackrel{˙}{m}$ denotes a mass flow rate into the pipe and the subscript denotes the port at which its value is defined. If the pipe is given a radially compliant wall—that is, if the Pipe wall specification block parameter is changed to `Flexible`—the thermal liquid mass contained within its bounds is free to vary, in a measure always directly proportional to the volume of the pipe:

`${\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}={\rho }_{\text{I}}\stackrel{˙}{V},$`

where ρ is the thermal density inside the pipe volume (subscript `I`), denoted V. If in addition the thermal liquid is made compressible—if the Fluid dynamic compressibility block parameter is changed to `On`—its mass within the pipe must change with pressure and temperature also. This dependence is captured by the bulk modulus and thermal expansion coefficient of the thermal liquid:

`${\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}={\rho }_{\text{I}}\stackrel{˙}{V}+{\rho }_{\text{I}}V\left(\frac{{\stackrel{˙}{p}}_{\text{I}}}{{\beta }_{\text{I}}}-{\alpha }_{\text{I}}{\stackrel{˙}{T}}_{\text{I}}\right),$`

where p and T denote the pressure and temperature within the pipe volume and β and α denote the bulk modulus and thermal expansion coefficient (as provided to the thermal liquid network by the Thermal Liquid Settings (TL) or Thermal Liquid Properties (TL) block).

Momentum Balance

The thermal liquid flow is subjected to different forces as it traverses the pipe. These arise due to static pressure at the ports, viscous friction along the pipe wall, and gravity on the volume of the pipe. The inertial force on the fluid is by default ignored, a suitable approximation at the large time scales over which changes to flow typically occur. The pipe is then treated as a quasi-steady component and its momentum balance—expressed as a pressure difference between its ports—becomes:

`${p}_{\text{A}}-{p}_{\text{B}}={p}_{\text{F,A}}-{p}_{\text{F,B}}+{\rho }_{\text{I}}g\Delta z,$`

where p is the pressure at a port and pF the pressure loss due to friction in half of the pipe volume; g is the gravitational acceleration and Δz the rise in elevation from port A to port B. The subscripts denote a port (A or B) or the computational node corresponding to the internal fluid volume (I). Pressure is assumed in this simple case to vary linearly between the ports. Its value at the internal node—that used in the table lookup of ρI—is for this reason defined as the arithmetic mean of the pressures at the ports:

`${p}_{\text{I}}=\frac{{p}_{\text{A}}+{p}_{\text{B}}}{2}.$`

If the flow is treated as compressible—if the Fluid dynamic compressibility block parameter is changed to `On`—the pressure in the pipe can vary nonlinearly between the ports. Its value is no longer a simple arithmetic mean and it must be obtained explicitly by another means. To carry out this calculation, the momentum balance is split over two control volumes, one each for half of the pipe volume. Between port A and the internal node:

`${p}_{\text{A}}-{p}_{\text{I}}={p}_{\text{F,A}}+{\rho }_{\text{I}}g\frac{\Delta z}{2}.$`

Between port B and the internal node:

`${p}_{\text{B}}-{p}_{\text{I}}={p}_{\text{F,B}}-{\rho }_{\text{I}}g\frac{\Delta z}{2}.$`

If, in addition, the inertia of the fluid is factored into the calculations—that is, if the Fluid inertia block parameter is set to `On`—then changes to flow momentum are no longer assumed to be instantaneous. The transient phase between old and new steady states becomes gradual, with a short but nonzero time scale that depends in part on the system modeled. The momentum balance becomes, in the control volume adjacent to port A:

`${p}_{\text{A}}-{p}_{\text{I}}={p}_{\text{F,A}}+{\rho }_{\text{I}}g\frac{\Delta z}{2}+\frac{{\stackrel{¨}{m}}_{\text{A}}}{S}\frac{L}{2},$`

where L is the length of the pipe and S is the cross-sectional area of the flow through the same. Reversing the sign of the elevation term gives for the control volume adjacent to port B:

`${p}_{\text{B}}-{p}_{\text{I}}={p}_{\text{F,B}}-{\rho }_{\text{I}}g\frac{\Delta z}{2}+\frac{{\stackrel{¨}{m}}_{\text{B}}}{S}\frac{L}{2},$`

Frictional Pressure Loss at the Pipe Wall

The calculation of the major pressure loss (due to friction in the pipe) varies with the viscous friction parameterization. For all parameterizations but `Nominal pressure drop vs. nominal mass flow rate`, the calculation is based on the Darcy-Weisbach equation:

`${p}_{\text{F},j}={f}_{\text{D},j}\frac{1}{4}\frac{{L}_{\text{E}}{\stackrel{˙}{m}}_{j}|{\stackrel{˙}{m}}_{j}|}{{\rho }_{\text{I}}D{S}^{2}},$`

where fD is the Darcy friction factor and the subscript j denotes the pipe half—that adjacent to port A or to port B. LE is the effective length of the pipe and D is the hydraulic diameter of the same. The effective pipe length is as the sum of the true pipe length and the aggregate equivalent length of all local resistances (those due to elbows, unions, fittings, and other local sources of friction).

When the flow is laminar, the friction factor (for a given pipe geometry) is a function of the Reynolds number alone:

`${f}_{\text{D},j}=\frac{\lambda }{{\text{Re}}_{j}},$`

where λ is the shape factor of the pipe, an empirical constant used to encode the effect of pipe geometry on the major pressure loss; its value is `64` in circular pipes and `48``96` in noncircular ones. The Reynolds number at port k is defined as:

`${\text{Re}}_{j}=\frac{{\stackrel{˙}{m}}_{j}D}{{\mu }_{\text{I}}S},$`

where μ is the dynamic viscosity obtained from the Thermal Liquid Settings (TL) or Thermal Liquid Properties (TL) block. The actual pressure loss calculation in the laminar flow regime is carried out as:

`${p}_{\text{F},j}=\frac{1}{4}\frac{\lambda {\mu }_{\text{I}}{L}_{\text{E}}{\stackrel{˙}{m}}_{j}}{{\rho }_{\text{I}}{D}^{2}S},$`

When the flow is turbulent, the friction factor is a function also of the pipe diameter and surface roughness. If the viscous friction parameterization is set to `Haaland correlation`, the friction factor is calculated from the empirical expression:

`$\frac{1}{\sqrt{{f}_{\text{D},j}}}=-1.8\text{log}\left[{\left(\frac{ϵ}{D}}{3.7}\right)}^{1.11}+\frac{6.9}{{\text{Re}}_{j}}\right],$`

where ε is the absolute roughness of the pipe, a measure of the height of the bumps at the pipe-fluid interface; typical roughness values range from 0.0015 mm for certain plastic and glass tubes to 3 mm for larger concrete pipes. If the viscous friction parameterization is set to ```Tabulated data – Darcy friction factor vs. Reynolds number```, the friction factor is obtained from the tabulated data as a function of the Reynolds number:

`${f}_{\text{D}}=f\left(\text{Re}\right).$`

If the viscous friction parameterization is set to ```Nominal pressure drop vs. nominal mass flow rate```, the major pressure loss is calculated for each pipe half from the expression:

`${p}_{\text{F},j}=\frac{1}{2}{K}_{\text{p}}{\stackrel{˙}{m}}_{j}\sqrt{{\stackrel{˙}{m}}_{j}^{2}+{\stackrel{˙}{m}}_{\text{Th}}^{2}}$`

where ${\stackrel{˙}{m}}_{\text{Th}}$ is a threshold mass flow rate, a small value specified in the block dialog box that is used for numerical smoothing purposes; Kp is a pressure loss coefficient, computed for rigid pipes as:

`${K}_{\text{p}}=\frac{{p}_{\text{F,N}}}{{\stackrel{˙}{m}}_{\text{N}}^{2}},$`

where the subscript `N` denotes a value specified at some nominal operating conditions. The nominal pressure and mass flow rate can be specified as scalars (in which case the pressure loss coefficient is fixed throughout simulation) or as a vector (in which case the pressure loss coefficient is determined as a variable, by interpolation or extrapolation of the tabulated data). The pressure loss coefficient is redefined for flexible pipes (with a slight change in physical units) as:

`${K}_{\text{p}}=\frac{{p}_{\text{F,N}}}{{\stackrel{˙}{m}}_{\text{N}}^{2}}{D}_{\text{N}}.$`

Hydraulic Diameter of the Pipe Wall

The hydraulic diameter is a measure of width for pipes whose cross sections may not be circular. Note that only rigid pipes are allowed to be noncircular in cross section. The diameter of a flexible pipe can vary during simulation as a function of pressure, with such changes assumed to be uniform throughout the length of the pipe. The deformation of the pipe proceeds at a rate set in part by a viscoelastic time constant:

`$\stackrel{˙}{D}=\frac{{D}_{\text{S}}-{D}_{\text{N}}}{\tau },$`

where τ is the time constant and the subscripts `S` and `N` denote the values at steady-state and nominal conditions, respectively. The nominal value gives the diameter at zero gauge pressure (when the pressure in the component is equal to atmospheric pressure). The steady-state value gives the diameter at the actual gauge pressure after the transient response has ceased:

`${D}_{\text{S}}={K}_{c}\frac{p-{p}_{\text{0}}}{\tau },$`

where Kc is the elastic compliance of the pipe wall, a number indicating the extent to which a change in pressure affects the diameter of the pipe. This parameter can be calculated if necessary from other elastic properties of the wall:

`${K}_{\text{c}}=\frac{{D}_{\text{0,Int}}}{E}\left(\frac{{D}_{\text{0,Ext}}^{2}+{D}_{\text{0,Int}}^{2}}{{D}_{\text{0,Ext}}^{2}-{D}_{\text{0,Int}}^{2}}+\nu \right),$`

where E and ν are the modulus of elasticity and Poisson ratio of the pipe wall material. The subscript `0` denotes an initial value, corresponding to the conditions in the model at the start of simulation (not to be confused with nominal conditions at which DN is specified). The subscripts `Int` and `Ext` refer to the internal and external circumferences of the pipe wall.

The nominal pipe diameter (used in the calculation of the pipe deformation rate) is computed as:

`${D}_{\text{N}}=\sqrt{\frac{4S}{\pi }},$`

where S is the specified cross-sectional area of the pipe (a nominal value in pipes that are treated as flexible).

Energy Balance

The energy of the fluid in the pipe can change by a variety of processes. These include advection and conduction through the ends of the pipe (thermal liquid ports A and B), convection at the pipe-fluid interface (thermal port H), and, in pipes that are set at an angle, longitudinal changes in elevation. Expressing the energy balance in terms of the energy accumulation rate in the pipe gives:

`$\stackrel{.}{{E}_{\text{I}}}={\varphi }_{\text{A}}+{\varphi }_{\text{B}}+{\varphi }_{\text{H}}-\stackrel{.}{{m}_{\text{I}}}g\Delta z,$`

where $\stackrel{˙}{E}$ is the energy accumulation rate and ϕ is the energy flow rate through a port—smoothed and upwinded in the thermal liquid ports, as described in Energy Flows in Thermal Liquid Networks. As in the mass and momentum calculations, the subscript I denotes a value defined at the internal computational node. The mass flow rate in the potential energy term is the average of those established at the thermal liquid ports:

`${\stackrel{˙}{m}}_{\text{I}}=\frac{{\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}}{2}.$`

The total energy of the internal fluid volume is defined in terms of the specific internal energy as:

`${E}_{\text{I}}={\rho }_{\text{I}}{u}_{\text{I}}V,$`

where u is the specific internal energy of the fluid, obtained as a function of temperature and pressure from the Thermal Liquid Settings (TL) or Thermal Liquid Properties (TL) block, and V is the internal volume of the pipe. If the flow is treated as compressible—if the Flow dynamic compressibility block parameter is set to `On`—then the energy accumulation rate in the pipe is computed as:

`${\stackrel{˙}{E}}_{\text{I}}={\rho }_{\text{I}}V{\left(\frac{du}{dp}\frac{dp}{dt}+\frac{du}{dT}\frac{dT}{dt}\right)}_{\text{I}},$`

If in addition the pipe is given a compliant wall—if the Pipe wall specification block parameter is set to `Flexible`—then the volume of thermal liquid within its bounds is free to vary. The energy accumulation rate becomes:

`${\stackrel{˙}{E}}_{\text{I}}={\rho }_{\text{I}}V{\left(\frac{du}{dp}\frac{dp}{dt}+\frac{du}{dT}\frac{dT}{dt}\right)}_{\text{I}}+\left({\rho }_{\text{I}}{u}_{\text{I}}+{p}_{\text{I}}\right){\left(\frac{dV}{dt}\right)}_{\text{I}},$`

Convective Heat Transfer at the Pipe Wall

The heat flow rate between the thermal liquid and the pipe wall is assumed to result from a convective exchange and a purely conductive exchange:

`${\varphi }_{\text{H}}={Q}_{\text{Conv}}+{Q}_{\text{Cond}}.$`

The heat flow rate due to conduction is computed as:

`${Q}_{\text{Cond}}=\frac{{k}_{\text{I}}{S}_{\text{H}}}{D}\left({T}_{\text{H}}-{T}_{\text{I}}\right),$`

where k is the thermal conductivity of the thermal liquid and SH is the surface area of the pipe wall (the product of the perimeter and length of the pipe, not to be confused with the cross-sectional area of the same). The subscripts `H` and `I` denote the pipe wall and the internal fluid volume, respectively.

The heat flow rate due to convection is computed as:

`${Q}_{\text{Conv}}={c}_{\text{p,Avg}}|{\stackrel{˙}{m}}_{\text{Avg}}|\left({T}_{\text{H}}-{T}_{\text{In}}\right)\left[1-\text{exp}\left(-\frac{h{S}_{\text{H}}}{{c}_{\text{p,Avg}}|{\stackrel{˙}{m}}_{\text{Avg}}|}\right)\right],$`

where cp is the specific heat of the thermal liquid, h is the heat transfer coefficient of the pipe. The subscript In denotes the pipe inlet (port A or B depending on flow direction). Parameters with the subscript Avg are evaluated at the average temperature of the pipe. This expression is based on the assumption that temperature varies exponentially between the ends of the pipe.

For all heat transfer parameterizations but ```Nominal temperature differential vs. nominal mass flow rate```, the heat transfer coefficient is computed from the expression:

`$h=\frac{{\text{Nu}}_{\text{Avg}}{k}_{\text{Avg}}}{D},$`

where Nu is the Nusselt number and k the thermal conductivity in the pipe, both obtained at the average temperature inside it. The Nusselt number calculation varies with the parameterization selected:

• `Gnielinski correlation`:

In the turbulent regime:

where Pr is the Prandtl number. In the laminar flow regime, in which the correlation does not apply, the Nusselt number is obtained as a constant (denoted NuL) from the Nusselt number for laminar flow heat transfer block parameter:

`${\text{Nu}}_{\text{Avg}}={\text{Nu}}_{\text{L}},$`

• `Dittus-Boelter correlation`:

In the turbulent regime:

`${\text{Nu}}_{\text{Avg}}=a{\text{Re}}_{\text{Avg}}^{b}{\text{Pr}}_{\text{Avg}}^{c},$`

where a, b, and c are empirical constants specific to the system considered. The default values specified in the block are those used in the exact form of the Dittus-Boelter expression for a fluid being warmed by the pipe wall:

`${\text{Nu}}_{\text{Avg}}=0.023{\text{Re}}_{\text{Avg}}^{0.8}{\text{Pr}}_{\text{Avg}}^{0.4}.$`

As with the Gnielinski correlation, in the laminar flow regime, in which the correlation does not apply, the Nusselt number is obtained as a constant (denoted NuL) from the Nusselt number for laminar flow heat transfer block parameter.

• ```Tabulated data - Colburn factor vs. Reynolds number```:

In all flow regimes:

`${\text{Nu}}_{\text{Avg}}={\text{J}}_{\text{M,Avg}}\left({\text{Re}}_{\text{Avg}}\right){\text{Re}}_{\text{Avg}}{\text{Pr}}_{\text{Avg}}^{1/3}.$`

where JM is the Colburn-Chilton factor.

• ```Tabulated data - Nusselt number vs. Reynolds number & Prandtl number```:

In all flow regimes:

`${\text{Nu}}_{\text{Avg}}=\text{Nu}\left({\text{Re}}_{\text{Avg}},{\text{Pr}}_{\text{Avg}}\right).$`

The calculations differ slightly in the case of the ```Nominal temperature difference vs. nominal mass flow rate``` parameterization. In the laminar flow regime, the heat transfer coefficient is the same constant specified in other parameterizations (Nusselt number for laminar flow heat transfer block parameter). In the turbulent flow regime, it is calculated as a function of mass flow rate, with the proportionality between the two fixed by a form of the Dittus-Boelter correlation:

`$\text{Nu}\propto {\text{Re}}^{0.8},$`

or:

`$\frac{hD}{k}\propto {\left(\frac{\stackrel{˙}{m}D}{S\mu }\right)}^{0.8}$`

Rearranging terms:

`$h={K}_{\text{H,Avg}}\frac{{\stackrel{˙}{m}}_{\text{Avg}}^{0.8}}{{D}^{1.8}},$`

where KH is a proportionality constant created by lumping all parameters but those retained in the final expression (with the fluid properties defined at the average temperature in the pipe). The constant is computed from nominal values obtained for h, D, and $\stackrel{˙}{m}$ as:

`${K}_{\text{H,Avg}}=\frac{{h}_{\text{N}}{D}_{\text{N}}^{1.8}}{{\stackrel{˙}{m}}_{\text{N}}^{0.8}},$`

The heat transfer coefficient for the ```Nominal temperature difference vs. nominal mass flow rate``` parameterization is therefore:

`$h=\frac{{h}_{\text{N}}{D}_{\text{N}}^{1.8}}{{\stackrel{˙}{m}}_{\text{N}}^{0.8}}\frac{{\stackrel{˙}{m}}_{\text{I}}^{0.8}}{{D}^{1.8}},$`

or, in the simpler case of pipe treated as rigid (and therefore assumed to be constant in diameter):

`${h}_{\text{I}}=\frac{{h}_{\text{N}}}{{\stackrel{˙}{m}}_{\text{N}}^{0.8}}{\stackrel{˙}{m}}_{\text{I}}^{0.8}.$`

The nominal mass flow rate is obtained from the tabulated data specified via the Nominal mass flow rate block parameter. The nominal heat transfer coefficient is calculated from various nominal parameters as:

where cp is the specific heat at constant pressure and the subscripts `H`, `In`, and `Out` denote the wall, the inlet (whichever of the thermal liquid ports happens to be it at a given moment), and the outlet. The nominal surface area of the pipe wall (S{H,N}) is computed as the product of the pipe circumference and the pipe length:

`${S}_{\text{H,N}}=\frac{4S}{D}L,$`

The hydraulic diameter (D) is a constant if the pipe is rigid but a function of pressure if the pipe is flexible. Its value is obtained from the Hydraulic diameter block parameter if the Pipe wall specification parameter is set to `Rigid` and computed from the Nominal cross-sectional area parameter otherwise, giving, for the flexible pipe:

Ports

Input

expand all

Control signal with which to set the instantaneous elevation difference between the thermal liquid ports. Depending on the blocks used to generate the signal, the elevation difference can be constant or it can vary with time. If the port is left unconnected, the elevation difference is fixed at zero during simulation.

Conserving

expand all

Opening through which the thermal liquid flows into or out of the pipe. Ports A and B can each function as either inlet or outlet. Thermal conduction is allowed between the thermal liquid ports and the fluid internal to the pipe (though its impact is typically relevant only at near zero flow rates).

Opening through which the thermal liquid flows into or out of the pipe. Ports A and B can each function as either inlet or outlet. Thermal conduction is allowed between the thermal liquid ports and the fluid internal to the pipe (though its impact is typically relevant only at near zero flow rates).

Thermal boundary between the fluid volume and the pipe wall. Use this port to capture heat exchanges of various kinds—for example, conductive, convective, or radiative—between the fluid and the environment external to the pipe (taking into account the thermal resistance of the wall when it is significant).

Parameters

expand all

Configuration

Option to capture variations in density due to those in pressure. Dynamic compressibility lends the block to more realistic, though also more computationally demanding, simulations. Select `Off` if simulation speed is of concern, as it generally is in real-time simulation, and if dynamic compressibility is of little consequence in the results of the model.

Option to capture the resistance of the fluid to changes in its state of motion, for example in systems prone to water hammer. Fluid inertia lends the block to more realistic, though also more computationally demanding, simulations. Select `Off` if simulation speed is of concern, as it generally is in real-time simulation, and if fluid inertia is of little consequence in the results of the model.

Dependencies

This parameter is active when the Fluid dynamic compressibility block parameter is set to `On`.

Number of lengths into which to discretize the pipe. Each length corresponds to a fluid volume with a computational node—the point in a component at which pressure and temperature are evaluated during simulation. Increase the number of pipe segments to more accurately capture the lengthwise distributions of pressure and temperature, for example in the simulation of water hammer, in which such distributions matter.

Dependencies

This parameter is active when the Fluid dynamic compressibility block parameter is set to `On`.

Sum of the lengths of the segments comprising the pipe.

Area of the inner circumference of the cross section of the pipe (in the undeformed state if modeled with a compliant wall).

Option to capture the radial compliance of the pipe wall. The default setting of `On` corresponds to a flexible tube whose wall expands and contracts as a function of pressure. Select `Off` if simulation speed is of concern, as it generally is in real-time simulation, and if wall compliance is of little consequence in (or uncharacteristic of) the model.

Dependencies

This parameter is active when the Fluid dynamic compressibility block parameter is set to `On`.

Ratio of the opening area of the pipe to the inner perimeter of (the cross-section of) the same. This parameter gives a general measure of width for pipes with noncircular cross sections.

Dependencies

This parameter is active when the Fluid dynamic compressibility block parameter is set to `Off` or when it is set to `On` but the Pipe wall specification block parameter is set to `Rigid`.

Change in the elevation of the pipe in the direction of port A to port B. This parameter allows for the calculation of the pressure change due to elevation in the pipe. The default value of `0` corresponds to a pipe laid flat.

Dependencies

This parameter is exposed in the block dialog box when the block variant is set to `Constant elevation`. Change the block variant to `Variable elevation` if necessary, for example to capture the tilting of the pipe in during simulation.

Value of the gravitational acceleration (g) at the mean elevation of the pipe. Any changes in elevation are assumed to be sufficiently small that the pull of gravity is approximately constant.

Measure of the radial deformation induced in a pipe by a unit change in the pressure within it relative to its surroundings. This parameter is a property of the material of which the pipe wall is made.

Dependencies

This parameter is active when the Pipe wall specification block parameter is set to `Flexible`.

Characteristic time scale of the elastic deformations produced on the pipe wall. This parameter gives a rough measure of the time needed for a pipe disturbed by a pressure change to reach a new steady-state diameter.

Dependencies

This parameter is active when the Pipe wall specification block parameter is set to `Flexible`.

Viscous Friction

Method by which to capture the pressure loss in the pipe due to friction against the wall. The calculation can be based on an empirical correlation (that of Haaland) or on a tabulated function (providing either the pressure drop or the Darcy friction factor).

Minor pressure loss in the pipe expressed as a length. This parameter serves to adjust the effective length of the pipe and from it to calculate the total pressure loss between the ports.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to `Haaland correlation`.

Characteristic height of the microscopic protrusions on the inner surface of the pipe. This parameter serves to calculate the pressure loss due to friction against the wall of the pipe.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to `Haaland correlation`.

Empirical measure of the effects of geometry on the pressure losses due to friction. Typical values range from `48` to `96`. The default value, `64`, corresponds to a pipe of circular cross section.

Reynolds number below which the flow is laminar. Above this threshold, the flow transitions to turbulent, reaching the true turbulent regime at the Turbulent flow lower Reynolds number limit setting.

Reynolds number above which the flow is turbulent. Below this threshold, the flow gradually transitions to laminar, reaching the true laminar regime at the Laminar flow upper Reynolds number limit setting.

Mass flow rate, at some chosen operating points, from which to calculate the pressure loss coefficient of the pipe. This parameter can be scalar or vector, with a scalar corresponding to a single operating point and a vector to a multitude of operating points. The MATLAB `mldivide` function is used to solve for the pressure loss coefficient if a vector is specified. All nominal parameters in the Viscous Friction tab must have the same size.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to ```Nominal pressure drop vs. nominal mass flow rate```.

Pressure drop, at some chosen operating points, from which to calculate the pressure loss coefficient of the pipe. This parameter can be scalar or vector, with a scalar corresponding to a single operating point and a vector to a multitude of operating points. The MATLAB `mldivide` function is used to solve for the pressure loss coefficient if a vector is specified. All nominal parameters in the Viscous Friction tab must have the same size.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to ```Nominal pressure drop vs. nominal mass flow rate```.

Mass flow rate below which to apply numerical smoothing to the block calculations, a measure taken to prevent simulation errors due to discontinuities at zero flow.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to ```Nominal pressure drop vs. nominal mass flow rate```.

Reynolds number at which to tabulate the Darcy friction factor. This data serves to construct a one-way lookup table from which to calculate the Darcy friction factor and ultimately the pressure loss across the pipe. The vector must increase monotonically from left to right. This and the Darcy friction factor vector must have the same size.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to ```Tabulated data - Darcy friction factor vs. Reynolds number```.

Darcy friction factors at the tabulated values of the Reynolds number. This data serves to construct a one-way lookup table from which to calculate the Darcy friction factor and ultimately the pressure loss across the pipe. This and the Reynolds number must have the same size.

Dependencies

This parameter is active when the Viscous friction parameterization block parameter is set to ```Tabulated data - Darcy friction factor vs. Reynolds number```.

Heat Transfer

Method by which to calculate the convective heat transfer coefficient of the pipe. The calculation can be based on an empirical correlation (of which those of Gnielinski and Dittus-Boelter are options) or on a tabulated function (providing the temperature differential, the Colburn factor, or the Nusselt number).

Nusselt number for laminar pipe flows. This number serves to calculate the heat transfer coefficient between the pipe wall and the fluid within it. The default value of `3.66` corresponds to flow through a pipe with a circular cross section.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to `Gnielinski correlation`, ```Nominal temperature differential vs. nominal mass flow rate```, or ```Dittus-Boelter correlation```.

Mass flow rates, at some chosen operating point, from which to calculate the heat transfer coefficient of the pipe. This parameter can be scalar or vector, with a scalar corresponding to a single operating point and a vector to a multitude of operating points. The MATLAB `mldivide` function is used to solve for the heat transfer coefficient if vector a vector is specified. All nominal parameters in the Heat Transfer tab must have the same size.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Nominal temperature differential vs. nominal mass flow rate```.

Pipe entrance temperatures, at some chosen operating points, from which to calculate the heat transfer coefficient of the pipe. This parameter can be scalar or vector, with a scalar corresponding to a single operating point and a vector to a multitude of operating points. The MATLAB `mldivide` function is used to solve for the heat transfer coefficient if a vector is specified. All nominal parameters in the Heat Transfer tab must have the same size.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Nominal temperature differential vs. nominal mass flow rate```.

Pipe exit temperatures, at some chosen operating points, from which to calculate the heat transfer coefficient of the pipe. This parameter can be scalar or vector, with a scalar corresponding to a single operating point and a vector to a multitude of operating points. The MATLAB `mldivide` function is used to solve for the heat transfer coefficient if a vector is specified. All nominal parameters in the Heat Transfer tab must have the same size.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Nominal temperature differential vs. nominal mass flow rate```.

Pipe entrance pressures, at some chosen operating points, from which to calculate the heat transfer coefficient of the pipe. This parameter can be scalar or vector, with a scalar corresponding to a single operating point and a vector to a multitude of operating points. The MATLAB `mldivide` function is used to solve for the heat transfer coefficient if a vector is specified. All nominal parameters in the Heat Transfer tab must have the same size.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Nominal temperature differential vs. nominal mass flow rate```.

Pipe wall temperatures, at some chosen operating points, from which to calculate the heat transfer coefficient of the pipe. This parameter can be scalar or vector, with a scalar corresponding to a single operating point and a vector to a multitude of operating points. The MATLAB `mldivide` function is used to solve for the heat transfer coefficient if a vector is specified. All nominal parameters in the Heat Transfer tab must have the same size.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Nominal temperature differential vs. nominal mass flow rate```.

Empirical constant a to use in the Dittus-Boelter correlation. The correlation gives the value of the Nusselt number from which to calculate the heat transfer coefficient in turbulent pipe flows. The default value is that most often associated with this correlation.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to `Dittus-Boelter correlation`.

Empirical constant c to use in the Dittus-Boelter correlation. The correlation gives the value of the Nusselt number from which to calculate the heat transfer coefficient in turbulent pipe flows. The default value is that most often associated with this correlation.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to `Dittus-Boelter correlation`.

Empirical constant c to use in the Dittus-Boelter correlation. The correlation gives the value of the Nusselt number from which to calculate the heat transfer coefficient in turbulent pipe flows. The default value is that most often associated with this correlation when the fluid is being warmed.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to `Dittus-Boelter correlation`.

Reynolds numbers at which to tabulate the Colburn factor. This data serves to construct a one-way lookup table from which to calculate the Colburn factor and ultimately the heat transfer coefficient. The vector must increase monotonically from left to right. This and the Colburn number vector must have the same size.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Tabulated data - Colburn factor vs. Reynolds number```.

Colburn factors at the tabulated values of the Reynolds number. This data serves to construct a one-way lookup table from which to calculate the Colburn factor and ultimately the heat transfer coefficient. This and the Reynolds number vector must have the same size.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Tabulated data - Colburn factor vs. Reynolds number```.

Reynolds numbers at which to tabulate the Nusselt number. This data serves to construct a two-way lookup table from which to calculate the Nusselt number and ultimately the heat transfer coefficient. The vector must increase monotonically from left to right. This vector must be equal in size to the number of rows in the Nusselt number table.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Tabulated data - Nusselt number vs. Reynolds number & Prandtl number```.

Prandtl numbers at which to tabulate the Nusselt number. This data serves to construct a two-way lookup table from which to calculate the Nusselt number and ultimately the heat transfer coefficient. The vector must increase monotonically from left to right. This vector must be equal in size to the number of columns in the Nusselt number table.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Tabulated data - Nusselt number vs. Reynolds number & Prandtl number```.

Nusselt numbers at the tabulated values of the Reynolds and Prandtl numbers. This data serves to construct a two-way lookup table from which to calculate the Nusselt number and ultimately the heat transfer coefficient.

The Reynolds number changes from row to row from top to bottom. The Prandlt number changes from column to column from left to right. The number of rows must be equal to the length of the Reynolds number vector and the number of columns to the length of the Prandtl number vector.

Dependencies

This parameter is active when the Heat transfer parameterization block parameter is set to ```Tabulated data - Nusselt number vs. Reynolds number & Prandtl number```.

Initial Conditions

Absolute temperature in the pipe at the start of simulation. This parameter can be a scalar, a vector of two elements, or a vector equal in size to the number of segments in the pipe. A scalar prescribes a constant temperature from end to end, a two-element vector a linear temperature gradient, and a vector of N values the individual temperatures of the various pipe segments.

Pressure in the pipe at the start of simulation. This parameter can be a scalar, a vector of two elements, or a vector equal in size to the number of segments in the pipe. A scalar prescribes a constant pressure from end to end, a two-element vector a linear pressure gradient, and a vector of N values the individual pressures of the various pipe segments.