Heat exchanger for systems with thermal liquid and controlled flows

**Library:**Simscape / Fluids / Fluid Network Interfaces / Heat Exchangers

The Heat Exchanger (TL) block models the complementary cooling and heating of fluids held briefly in thermal contact across a thin conductive wall. At least one of the fluids is single phase—a thermal liquid. This fluid cannot switch phase and so, as latent heat is never released, the exchange is strictly one of sensible heat. The second fluid is of a controlled sort—not part of a fluid domain but instead modeled with physical signals and a thermal port.

The heat transfer model depends on the choice of block variant. The block has two
variants: `E-NTU Model`

and ```
Simple
Model
```

. Right-click the block to open its context-sensitive menu and select **Simscape** > **Block Choices** to change variant.

`E-NTU Model`

The default variant. Its heat transfer model derives from the Effectiveness-NTU method. Heat transfer in the steady state then proceeds at a fraction of the ideal rate which the flows, if kept each at its inlet temperature, and if cleared of every thermal resistance in between, could in theory support:

$${Q}_{\text{Act}}=\u03f5{Q}_{\text{Max}},$$

where *Q*_{Act} the actual heat
transfer rate, *Q*_{Max} is the ideal heat transfer
rate, and *ε* is the fraction of the ideal rate actually observed in a real
heat exchanger encumbered with losses. The fraction is the heat exchanger effectiveness, and
it is a function of the number of transfer units, or NTU, a measure of the ease with which
heat moves between flows, relative to the ease with which the flows absorb that heat:

$$NTU=\frac{1}{R{C}_{\text{Min}}},$$

where the fraction is the overall thermal conductance between the flows
and *C*_{Min} is the smallest of the heat capacity
rates from among the flows—that belonging to the flow least capable of absorbing heat. The
heat capacity rate of a flow depends on the specific heat of the fluid
(*c*_{p}) and on its mass flow rate through the
exchanger ($$\dot{m}$$):

$$C={c}_{\text{p}}\dot{m}.$$

The effectiveness depends also on the relative disposition of the flows, the number of passes between them, and the mixing condition for each. This dependence reflects in the effectiveness expression used, with different flow arrangements corresponding to different expressions. For a list of the effectiveness expressions, see the E-NTU Heat Transfer block.

Use the **Flow arrangement** block parameter to set how the flows
meet in the heat exchanger. The flows can run parallel to each other, counter to each
other, or across each other. They can also run in a pressurized shell, one through tubes
enclosed in the shell, the other around those same tubes. The figure shows an example. The
tube flow can make one pass through the shell flow (shown right) or, for greater exchanger
effectiveness, multiple passes (left).

Other flow arrangements are possible through a generic parameterization based on tabulated effectiveness data and requiring little detail about the heat exchanger. Flow arrangement, mixing condition, and number of shell or tube passes, if relevant to the heat exchanger, are assumed to manifest in the tabulated data.

Use the **Cross flow type** parameter to mix each of the flows, one
of the flows, or none of the flows. Mixing in this context is the lateral movement of
fluid in channels that have no internal barriers, normally guides, baffles, fins, or
walls. Such movement serves to even out temperature variations in the transverse plane.
Mixed flows have variable temperature in the longitudinal plane alone. Unmixed flows have
variable temperature in both the transverse and longitudinal planes. The figure shows a
mixed flow (**i**) and an unmixed flow (**ii**).

The distinction between mixed and unmixed flows is considered only in cross flow arrangements. There, longitudinal temperature variation in one fluid produces transverse temperature variation in the second fluid that mixing can even out. In counter and parallel flow arrangements, longitudinal temperature variation in one fluid produces longitudinal temperature variation in the second fluid and mixing, as it is of little effect here, is ignored.

Shell-and-tube exchangers with multiple passes (**iv.b**-**e** in the figure for 2, 3, and 4
passes) are most effective. Of exchangers with a single pass, those with counter flows
(**ii** are most effective and those with parallel flows
(**i**) are least.

Cross-flow exchangers are intermediate in effectiveness, with mixing condition playing
a factor. They are most effective when both flows are unmixed (**iii.a**) and least effective when both flows are mixed (**iii.b**). Mixing just the flow with the smallest heat capacity rate (**iii.c**) lowers the effectiveness more than mixing just the flow
with the largest heat capacity rate (**iii.d**).

The overall thermal resistance, *R*, is the sum of the local
resistances lining the heat transfer path. The local resistances arise from convection at
the surfaces of the wall, conduction through the wall, and, if the wall sides are fouled,
conduction through the layers of fouling. Expressed in order from the thermal liquid side
(subscript 1) to the controlled fluid side (subscript 2):

$$R=\frac{1}{{U}_{\text{1}}{A}_{\text{Th,1}}}+\frac{{F}_{\text{1}}}{{A}_{\text{Th,1}}}+{R}_{\text{W}}+\frac{{F}_{\text{2}}}{{A}_{\text{Th,2}}}+\frac{1}{{U}_{\text{2}}{A}_{\text{Th,2}}},$$

where *U* is the convective heat transfer coefficient,
*F* is the fouling factor, and
*A*_{Th} is the heat transfer surface area, each
for the flow indicated in the subscript. *R*_{W} is
the thermal resistance of the wall.

The wall thermal resistance and fouling factors are simple constants obtained from
block parameters. The heat transfer coefficients are elaborate functions of fluid
properties, flow geometry, and wall friction, and derive from standard empirical
correlations between Reynolds, Nusselt, and Prandtl numbers. The correlations depend on
flow arrangement and mixing condition, and are detailed for each in the E-NTU Heat
Transfer block on which the `E-NTU Model`

variant is
based.

The `E-NTU Model`

variant is a composite component built from
simpler blocks. A Heat Exchanger Interface (TL) block
models the thermal liquid flow. Physical signals for the heat capacity rate and heat
transfer coefficient, along with a thermal port for temperature, capture the controlled
flow. An E-NTU Heat Transfer block models the
heat exchanged across the wall between the flows. The figure shows the block connections
for the `E-NTU Model`

block variant.

`Simple Model`

The alternative variant. Its heat transfer model depends on the concept of
*specific dissipation*, a measure of the heat transfer rate observed
when thermal liquid and controlled fluid inlet temperatures differ by one degree. Its
product with the inlet temperature difference gives the expected heat transfer rate:

$$Q=\xi ({T}_{\text{In,1}}-{T}_{\text{In,2}}),D$$

where *ξ* is specific dissipation and
*T*_{In} is inlet temperature for thermal liquid
(subscript `1`

) or controlled fluid (subscript `2`

). The
specific dissipation is a tabulated function of the mass flow rates into the exchanger
through the thermal liquid and controlled fluid inlets:

$$\xi =f({\dot{m}}_{\text{In,1}},{\dot{m}}_{\text{In,2}}),$$

To accommodate reverse flows, the tabulated data can extend over positive and negative flow rates, in which case the inlets can also be thought of as outlets. The data normally derives from measurement of heat transfer rate against temperature in a real prototype:

$$\xi =\frac{Q}{{T}_{\text{In,1}}-{T}_{\text{In,2}}}.$$

The heat transfer model, as it relies almost entirely on tabulated data, and as that data normally derives from experiment, requires little detail about the exchanger. Flow arrangement, mixing condition, and number of shell or tube passes, if relevant to the heat exchanger modeled, are assumed to manifest entirely in the tabulated data.

The `Simple Model`

variant is a composite component. A
Simple Heat Exchanger
Interface (TL) block models the thermal liquid flow. Physical signals
for the heat transfer coefficient and mass flow rate, along with a thermal port for
temperature, capture the controlled flow. A Specific Dissipation Heat
Transfer block models the heat exchanged across the wall between the
flows.

E-NTU Heat Transfer | Heat Exchanger Interface (TL) | Simple Heat Exchanger Interface (TL) | Specific Dissipation Heat Transfer