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Heat exchanger for systems with two thermal liquid flows

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

The Heat Exchanger (TL-TL) block models the complementary cooling and heating of fluids held briefly in thermal contact across a thin conductive wall. The wall can store heat in its bounds, adding to the heat transfer a slight transient delay that scales in proportion to its thermal mass. The fluids are single phase—each a thermal liquid. Neither fluid can switch phase and so, as latent heat is never released, the exchange is strictly one of sensible heat.

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** 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 thermal liquid side 1 to thermal liquid side 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 wall is more than a thermal resistance for heat to pass through. It is also a thermal mass and, like the flows it divides, it can store heat in its bounds. The storage slows the transition between steady states so that a thermal perturbation on one side does not promptly manifest on the side across. The lag persists for the short time that it takes the heat flow rates from the two sides to balance each other. That time interval scales with the thermal mass of the wall:

$${C}_{\text{Q,W}}={c}_{\text{p,W}}{M}_{\text{W}},$$

where is the *c*_{p,W} is the specific
heat capacity and *M*_{W} the inertial mass
of the wall. Their product gives the energy required to raise wall temperature
by one degree. Use the **Wall thermal mass** parameter to
specify that product. The parameter is active when the **Wall thermal
dynamics** setting is `On`

.

Thermal mass is often negligible in low-pressure systems. Low pressure affords a thin wall with a transient response so fast that, on the time scale of the heat transfer, it is virtually instantaneous. The same is not true of high-pressure systems, common in the production of ammonia by the Haber process, where pressure can break 200 atmospheres. To withstand the high pressure, the wall is often thicker, and, as its thermal mass is heftier, so its transient response is slower.

Set the **Wall thermal dynamics** parameter to
`Off`

to ignore the transient lag, cut the
differential variables that produce it, and, in reducing calculations, speed up
the rate of simulation. Leave it `On`

to capture the
transient lag where it has a measurable effect. Experiment with the setting if
necessary to determine whether to account for thermal mass. If simulation
results differ to a considerable degree, and if simulation speed is not a
factor, keep the setting `On`

.

The wall, if modeled with thermal mass, is considered in halves. One half sits on thermal liquid side 1 and the other half sits on thermal liquid side 2. The thermal mass divides evenly between the pair:

$${C}_{\text{Q,1}}={C}_{\text{Q,2}}=\frac{{C}_{\text{Q,W}}}{2}.$$

Energy is conserved in the wall. In the simple case of a wall half at steady state, heat gained from the fluid equals heat lost to the second half. The heat flows at the rate predicted by the E-NTU method for a wall without thermal mass. The rate is positive for heat flows directed from side 1 of the heat exchanger to side 2:

$${Q}_{\text{1}}=-{Q}_{\text{2}}=\u03f5{Q}_{\text{Max}}.$$

In the transient state, the wall is in the course of storing or losing heat, and heat gained by one half no longer equals that lost to the second half. The difference in the heat flow rates varies over time in proportion to the rate at which the wall stores or loses heat. For side 1 of the heat exchanger:

$${Q}_{\text{1}}=\u03f5{Q}_{\text{Max}}+{C}_{\text{Q,1}}{\dot{T}}_{\text{W,1}},$$

where $${\dot{T}}_{\text{W,1}}$$ is the rate of change in temperature in the wall half. Its product with the thermal mass of the wall half gives the rate at which heat accumulates there. That rate is positive when temperature rises and negative when it drops. The closer the rate is to zero the closer the wall is to steady state. For side 2 of the heat exchanger:$${Q}_{\text{2}}=-\u03f5{Q}_{\text{Max}}+{C}_{\text{Q,2}}{\dot{T}}_{\text{W,2}},$$

The `E-NTU Model`

variant is a composite component
built from simpler blocks. A Heat Exchanger Interface
(TL) block models the thermal liquid flow on side 1 of the
heat exchanger. Another models the thermal liquid flow on side 2. 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 1 and thermal liquid 2 inlet temperatures differ by
one degree. Its product with the inlet temperature difference gives the expected
heat transfer rate:

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

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

) or thermal liquid 2 (subscript
`2`

). The specific dissipation is a tabulated function of the
mass flow rates into the exchanger through the thermal liquid 1 and thermal liquid 2 inlets:

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

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.

See the Specific Dissipation Heat Transfer block for more detail on the heat transfer calculations.

The `Simple Model`

variant is a composite component.
A Simple Heat
Exchanger Interface (TL) block models the thermal liquid
flow on side 1 of the heat exchanger. Another models the thermal liquid flow on
side 2. A Specific
Dissipation Heat Transfer block captures the heat exchanged
across the wall between the flows.

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