Detailed heat transfer model between two general fluids

Fluid Network Interfaces/Heat Exchangers/Fundamental Components

The E-NTU Heat Transfer block models the heat exchange between two general fluids based on the standard Effectiveness-NTU method. The fluid thermal properties are specified explicitly through Simscape™ physical signals. Combine with the Heat Exchanger Interface (TL) block to model the pressure drop and temperature change between the inlet and outlet of a heat exchanger.

The block dialog box provides a choice of common heat exchanger configurations. These include concentric-pipe with parallel and counter flows, shell-and-tube with one or more shell passes, and cross-flow with mixed and unmixed flows. A generic configuration lets you model other heat exchangers based on tabular effectiveness data.

**Heat Exchanger Configurations**

The E-NTU model defines the heat transfer rate between fluids
1 and 2 in terms of an effectiveness parameter *ε*:

$$\begin{array}{cc}{Q}_{1}=-{Q}_{2}=\u03f5{Q}_{Max},& 0<\epsilon <1\end{array},$$

*Q*_{1}and*Q*_{2}are the heat transfer rates into fluid 1 and fluid 2.*Q*_{Max}is the maximum possible heat transfer rate between fluid 1 and fluid 2 at a given set of operating conditions.*ε*is the effectiveness parameter.

The maximum possible heat transfer rate between the two fluids is

$${Q}_{Max}={C}_{Min}\left({T}_{1,In}-{T}_{2,In}\right),$$

*C*_{Min}is the minimum value of the thermal capacity rate:$${C}_{Min}=min\left({\dot{m}}_{1}{C}_{p,1},{\dot{m}}_{2}{C}_{p,2}\right)$$

*T*_{1,In}and*T*_{2,In}are the inlet temperatures of fluid 1 and fluid 2.$${\dot{m}}_{1}$$ and $${\dot{m}}_{2}$$ are the mass flow rates of fluid 1 and fluid 2 into the heat exchanger volume through the inlet.

*c*_{p,1}and*c*_{p,2}are the specific heat coefficients at constant pressure of fluid 1 and fluid 2. The**Minimum fluid-wall heat transfer coefficient**parameter in the block dialog box sets a lower bound on the allowed values of the heat transfer coefficients.

The heat exchange effectiveness calculations depend on the flow
arrangement type selected in the block dialog box. For all but ```
Generic
— effectiveness table
```

, the block computes the
thermal exchange effectiveness through analytical expressions written
in terms of the number of transfer units (NTU) and thermal capacity
ratio. The number of transfer units is defined as

$$NTU=\frac{{U}_{Overall}{A}_{Heat}}{{C}_{Min}}=\frac{1}{{C}_{Min}{R}_{Overall}},$$

*NTU*is the number of transfer units.*U*_{Overall}is the overall heat transfer coefficient between fluid 1 and fluid 2.*R*_{Overall}is the overall thermal resistance between fluid 1 and fluid 2.*A*_{Heat}is aggregate area of the primary and secondary, or finned, heat transfer surfaces.

The thermal capacity ratio is defined as

$${C}_{rel}=\frac{{C}_{Min}}{{C}_{Max}}$$

*C*_{rel}is the thermal capacity ratio.

The overall heat transfer coefficient and thermal resistance used in the NTU calculation are functions of the heat transfer mechanisms at work. These mechanisms include convective heat transfer between the fluids and the heat exchanger interface and conduction through the interface wall [2]:

$${R}_{Overall}=\frac{1}{{U}_{Overall}{A}_{Heat}}=\frac{1}{{h}_{1}{A}_{Heat,1}}+{R}_{Foul,1}+{R}_{Wall}+{R}_{Foul,2}+\frac{1}{{h}_{2}{A}_{Heat,2}},$$

*h*_{1}and*h*_{2}are the heat transfer coefficients between fluid 1 and the interface wall and between fluid 2 and the interface wall.*A*_{Heat,1}and*A*_{Heat,2}are the heat transfer surface areas on the fluid-1 and fluid-2 sides.*R*_{Foul,1}and*R*_{Foul,2}are the fouling resistances on the fluid-1 and fluid-2 sides.*R*_{Wall}is the interface wall thermal resistance.

**Heat Transfer From Fluid 1 to Fluid 2**

The tables show some of the analytical expressions used to compute
the heat exchange effectiveness [1].
The parameter *N* refers to the number of shell passes
and the parameter *ε*_{1} to
the effectiveness for a single shell pass.

Concentric
Tubes | |

Counter Flow | $$\epsilon =\{\begin{array}{cc}\frac{1-\mathrm{exp}\left[-NTU\left(1-{C}_{rel}\right)\right]}{1-{C}_{rel}\mathrm{exp}\left[-NTU\left(1-{C}_{rel}\right)\right]},& \text{if}{C}_{rel}1\\ \frac{NTU}{1+NTU},& \text{if}{C}_{rel}=1\end{array}$$ |

Parallel Flow | $$\epsilon =\frac{1-\mathrm{exp}\left[-NTU\left(1+{C}_{rel}\right)\right]}{1+{C}_{rel}}$$ |

Shell
and Tube | |

One shell pass and two, four, or six tube passes | $${\epsilon}_{1}=\frac{2}{1+{C}_{rel}+\sqrt{1+{C}_{rel}{}^{2}}\frac{1+\mathrm{exp}\left(-NTU\sqrt{1+{C}_{rel}{}^{2}}\right)}{1-\mathrm{exp}\left(-NTU\sqrt{1+{C}_{rel}{}^{2}}\right)}}$$ |

| $$\epsilon =\frac{{\left[\left(1-{\epsilon}_{1}{C}_{rel}\right)/\left(1-{\epsilon}_{1}\right)\right]}^{N}-1}{{\left[\left(1-{\epsilon}_{1}{C}_{rel}\right)/\left(1-{\epsilon}_{1}\right)\right]}^{N}-{C}_{rel}}$$ |

Cross
Flow (Single Pass) | |

Both Fluids Unmixed | $$\epsilon =1-\mathrm{exp}\left(\frac{\mathrm{exp}\left(-{C}_{rel}NT{U}^{0.78}\right)-1}{{C}_{rel}NT{U}^{-0.22}}\right)$$ |

Both Fluids Mixed | $$\epsilon =\frac{1}{\frac{1}{1-exp\left(-NTU\right)}+\frac{{C}_{rel}}{1-\mathrm{exp}\left(-{C}_{rel}NTU\right)}-\frac{1}{NTU}}$$ |

C_{Max} mixed, C_{Min} unmixed | $$\epsilon =\frac{1}{{C}_{rel}}\left(1-\mathrm{exp}\left(-{C}_{rel}\left(1-\mathrm{exp}\left(-NTU\right)\right)\right)\right)$$ |

C_{Max} unmixed, C_{Min} mixed | $$\epsilon =1-\mathrm{exp}\left(-\frac{1}{{C}_{rel}}\left(1-\mathrm{exp}\left(-{C}_{rel}NTU\right)\right)\right)$$ |

The heat exchanger fluids do not change phase. They are always thermal liquids.

The heat exchanger is an adiabatic component. It does not transfer heat with its environment.

**Flow arrangement**Heat exchanger geometry. Common geometries that you can select include

`Parallel or counter flow`

,`Shell and tube`

, and`Cross flow`

. Select`Generic — effectiveness table`

to model other heat exchanger geometries based on tabular effectiveness data.In the

`Parallel or counter flow`

configuration, the relative flow directions of fluids 1 and 2 determine whether the heat exchanger is based on parallel or counter flows. The flow directions depend on the remainder of the Simscape Fluids™ model.**Number of shell passes**Number of times the fluid in the tubes traverses the length of the shell before exiting. A single shell pass corresponds to a single traversal of the shell length between the tube inlet and outlet. The figure shows one- and two-pass examples.

**One- and Two-Pass Configurations**This parameter is visible only when the

**Flow arrangement**parameter is set to`Shell and tube`

. The default value is`1`

, corresponding to a single shell pass.**Cross flow type**Fluid mixing configuration. The fluids can be mixed or unmixed. The block uses the mixing configuration to determine which empirical heat transfer correlations to use. This parameter is visible only when the

**Flow arrangement**parameter is set to`Cross flow`

. The default setting is`Both fluids mixed`

.**Number of heat transfer units vector, NTU***M*-element vector of NTU values at which to specify the effectiveness tabular data. The number of transfer units (NTU) is a dimensionless parameter defined aswhere:$$NTU=\frac{{A}_{s}U}{{C}_{min}},$$

*A*_{S}is the heat transfer surface area.*U*is the overall heat transfer coefficient.*C*_{min}is the smallest of the thermal capacity rates for the hot and cold fluids.

This parameter is visible only when the

**Flow Arrangement**parameter is set to`Generic — effectiveness table`

. The default vector is`[0.5, 1.0, 2.0, 3.0, 4.0]`

.**Thermal capacity ratio vector, CR***N*-element vector of thermal capacity ratios at which to specify the effectiveness tabular data. The thermal capacity ratio is the fractionwhere$${C}_{r}=\frac{{C}_{min}}{{C}_{max}},$$

*C*_{min}and*C*_{max}are the minimum and maximum thermal capacity rates. This parameter is visible only when the**Flow arrangement**parameter is set to`Generic — effectiveness table`

. The default vector is`[0.0, 0.25, 0.5, 0.75, 1.0]`

.**Effectiveness table, E(NTU, CR)***M*-by-*N*matrix with the heat exchanger effectiveness values. The matrix rows correspond to the different values specified in the**Number of heat transfer units vector, NTU**parameter. The matrix columns correspond to the values specified in the**Thermal capacity ratio vector, CR**parameter.This parameter is visible only when the

**Flow arrangement**parameter is set to`Generic — effectiveness table`

. The default table is a 6-by-5 matrix rangin in value from`0.30`

to`0.99`

.**Wall thermal resistance**Thermal resistance of the interface wall separating the two heat exchanger fluids. The block uses this parameter to compute the rate of heat transfer between the fluids. The default value is

`1.6e-4`

k/W.

**Heat transfer surface area**Aggregate surface area for heat transfer between the cold and hot fluids. The default value is

`0.01`

m^2.**Fouling factor**Empirical parameter used to quantify the increased thermal resistance due to dirt deposits on the heat transfer surface. The default value is

`1e-4`

m^2*K/W.**Minimum fluid-wall heat transfer coefficient**Smallest allowed value of the heat transfer coefficient. The heat transfer coefficients specified through physical signal ports HC1 and HC2 saturate at this value. The default value is

`5`

W/(m^2*K).The block uses the heat transfer coefficient to calculate the heat transfer rate between fluids 1 and 2 as described in Heat Transfer Rate.

H1 — Thermal conserving port associated with the inlet temperature of fluid 1

H2 — Thermal conserving port associated with the inlet temperature of fluid 2

C1 — Physical signal input port for the thermal capacity rate of fluid 1

C2 — Physical signal input port thermal capacity rate of fluid 2

HC1 — Physical signal input port for the heat transfer coefficient between fluid 1 and the interface wall

HC2 — Physical signal input port for the heat transfer coefficient between fluid 2 and the interface wall

[1] Holman, J. P. *Heat Transfer*. 9th
ed. New York, NY: McGraw Hill, 2002.

[2] Shah, R. K. and D. P. Sekulic. *Fundamentals
of Heat Exchanger Design*. Hoboken, NJ: John Wiley &
Sons, 2003.

Was this topic helpful?