E-NTU Heat Transfer

Detailed heat transfer model between two general fluids

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Libraries:
Simscape / Fluids / Heat Exchangers / Fundamental Components

Description

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) or Heat Exchanger Interface (G) blocks 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

Heat Transfer Rate

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

$$\begin{array}{cc}-Q_1=Q_2=ϵQ_{Max},& 0<\epsilon <1\end{array},$$

where:

• Q₁ and Q₂ 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

$$Q_{Max}=C_{Min}\left(T_{1,In}-T_{2,In}\right),$$

where:

• 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.

Heat Exchanger Effectiveness

The heat exchanger 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

$$NTU=\frac{U_{Overall}{A}_{Heat}}{C_{Min}}=\frac{1}{C_{Min}{R}_{Overall}},$$

where:

• 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

$$C_{rel}=\frac{C_{Min}}{C_{Max}}$$

where:

• 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]:

$$R_{Overall}=\frac{1}{U_{Overall}{A}_{Heat}}=\frac{1}{h_1{A}_{Heat,1}}+R_{Foul,1}+R_{Wall}+R_{Foul,2}+\frac{1}{h_2{A}_{Heat,2}},$$

where:

• h₁ and h₂ 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. The fouling resistance is equal to the fouling factor parameter divided by the heat transfer surface area.

• 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 ε₁ to the effectiveness for a single shell pass.

Concentric Tubes
Counter Flow	$$\epsilon =\{\begin{array}{cc}\frac{1-\exp \left[-NTU\left(1-C_{rel}\right)\right]}{1-C_{rel}\exp \left[-NTU\left(1-C_{rel}\right)\right]},& \text{if }{C}_{rel}<1\\ \frac{NTU}{1+NTU},& \text{if }{C}_{rel}=1\end{array}$$
Parallel Flow	$$\epsilon =\frac{1-\exp \left[-NTU\left(1+C_{rel}\right)\right]}{1+C_{rel}}$$
Shell and Tube
One shell pass and two, four, or six tube passes	$${\epsilon }_1=\frac{2}{1+C_{rel}+\sqrt{1+C_{rel}{}^2}\frac{1+\exp \left(-NTU\sqrt{1+C_{rel}{}^2}\right)}{1-\exp \left(-NTU\sqrt{1+C_{rel}{}^2}\right)}}$$
N Shell Passes and 2N, 4N, or 6N Tube Passes	$$\epsilon =\frac{{\left[\left(1-{\epsilon }_1{C}_{rel}\right)/\left(1-{\epsilon }_1\right)\right]}^N-1}{{\left[\left(1-{\epsilon }_1{C}_{rel}\right)/\left(1-{\epsilon }_1\right)\right]}^N-C_{rel}}$$
Cross Flow (Single Pass)
Both Fluids Unmixed	$$\epsilon =1-\exp \left(\frac{\exp \left(-C_{rel}NT{U}^{0.78}\right)-1}{C_{rel}NT{U}^{-0.22}}\right)$$
Both Fluids Mixed	$$\epsilon =\frac{1}{\frac{1}{1-exp\left(-NTU\right)}+\frac{C_{rel}}{1-\exp \left(-C_{rel}NTU\right)}-\frac{1}{NTU}}$$
CMax mixed, CMin unmixed	$$\epsilon =\frac{1}{C_{rel}}\left(1-\exp \left(-C_{rel}\left(1-\exp \left(-NTU\right)\right)\right)\right)$$
CMax unmixed, CMin mixed	$$\epsilon =1-\exp \left(-\frac{1}{C_{rel}}\left(1-\exp \left(-C_{rel}NTU\right)\right)\right)$$

Assumptions and Limitations

The flows are single-phase. The heat transfer is strictly one of sensible heat. The transfer is limited to interior of the exchanger, with the environment neither gaining heat from nor providing heat to the flows—the heat exchanger is an adiabatic component.

Ports

Input

C1 — Fluid 1 thermal capacity

physical signal

Physical signal input port for the thermal capacity rate of fluid 1. The thermal capacity rate is the mass flow rate multiplied by the specific heat coefficient for the fluid.

C2 — Fluid 2 thermal capacity

physical signal

Physical signal input port for the thermal capacity rate of fluid 2. The thermal capacity rate is the mass flow rate multiplied by the specific heat coefficient for the fluid.

HC1 — Fluid 1 heat transfer coefficient

physical signal

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

HC2 — Fluid 2 heat transfer coefficient

physical signal

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

Conserving

H1 — Fluid 1 thermal inlet temperature

thermal

Thermal conserving port associated with the inlet temperature of fluid 1.

H2 — Fluid 2 thermal inlet temperature

thermal

Thermal conserving port associated with the inlet temperature of fluid.

Parameters

Flow arrangement — Manner in which the flows align in the heat exchanger

Parallel or counter flow (default) | Shell and tube | Cross flow | Generic — effectiveness table

Heat exchanger geometry. 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.

Wall thermal resistance — Resistance of the wall to heat flow by thermal conduction

1.6e-4 K/W (default) | positive scalar

Thermal resistance of the interface wall separating the two heat exchanger fluids.

Number of shell passes — Number of times the flow traverses the shell before exiting

1 (default) | positive scalar

Number of times the flow traverses the shell before exiting.

Dependencies

To enable this parameter, set Flow arrangement to Shell and tube.

Cross flow type — Mixing condition in each of the flow channels

Both fluids mixed (default) | Both fluids unmixed | Controlled Fluid 1 mixed & Controlled Fluid 2 unmixed | Controlled Fluid 1 unmixed & Controlled Fluid 2 mixed

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. Mixed flow means that the fluid is free to move in the transverse direction as it travels along the flow path. Unmixed flow means that the fluid is restricted to travel only along the flow path. For example, a side with fins is considered an unmixed flow.

Dependencies

To enable this parameter, set Flow arrangement to Cross flow.

Number of heat transfer units vector, NTU — Number of transfer units at each breakpoint in lookup table for heat exchanger effectiveness

[.5, 1, 2, 3, 4, 5] (default) | vector of positive scalars

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 as

$$NTU=\frac{A_{s}U}{C_{min}},$$

where:

• 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.

Dependencies

To enable this parameter, set Flow arrangement to Generic — effectiveness table.

Thermal capacity ratio vector, CR — Thermal capacity ratio at each breakpoint in lookup table for heat exchanger effectiveness

[0, .25, .5, .75, 1] (default) | vector of positive scalars

N-element vector of thermal capacity ratios at which to specify the effectiveness tabular data. The thermal capacity ratio is the fraction

$$C_r=\frac{C_{min}}{C_{max}},$$

where C_min and C_max are the minimum and maximum thermal capacity rates.

Dependencies

To enable this parameter, set Flow arrangement to Generic — effectiveness table.

Effectiveness table, E(NTU,CR) — Heat exchanger effectiveness at each breakpoint in lookup table over the number of transfer units and thermal capacity ratio

[.3, .3, .3, .3, .3; .6, .55, .5, .47, .43; .85, .76, .68, .61, .55; .94, .83, .72, .65, .58; .98, .86, .75, .66, .58; .99, .86, .75, .66, .58] (default) | table of positive scalars

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.

Dependencies

To enable this parameter, set Flow arrangement to Generic — effectiveness table.

Controlled Fluid 1

Heat transfer surface area — Aggregate heat transfer surface area on the controlled fluid 1 side

0.01 m^2 (default) | positive scalar

Aggregate surface area on fluid 1 side for heat transfer between the cold and hot fluids.

Fouling factor — Measure of thermal resistance due to fouling deposits

1e-4 K*m^2/W (default) | positive scalar

Empirical parameter used to quantify the increased thermal resistance due to dirt deposits on the heat transfer surface.

Minimum fluid-wall heat transfer coefficient — Lower bound for the heat transfer coefficient

5 W/(m^2 * K) (default) | positive scalar

Smallest allowed value of the heat transfer coefficient. The heat transfer coefficient specified through physical signal ports HC1 saturates at this value. The block uses the heat transfer coefficient to calculate the heat transfer rate between fluids 1 and 2 as described in Heat Transfer Rate.

Controlled Fluid 2

Heat transfer surface area — Aggregate heat transfer surface area on the controlled fluid 2 side

0.01 m^2 (default) | positive scalar

Aggregate surface area on fluid 2 side for heat transfer between the cold and hot fluids.

Fouling factor — Measure of thermal resistance due to fouling deposits

1e-4 K*m^2/W (default) | positive scalar

Empirical parameter used to quantify the increased thermal resistance due to dirt deposits on the heat transfer surface.

Minimum fluid-wall heat transfer coefficient — Lower bound for the heat transfer coefficient

5 W/(m^2 * K) (default) | positive scalar

Smallest allowed value of the heat transfer coefficient. The heat transfer coefficient specified through physical signal ports HC2 saturates at this value. The block uses the heat transfer coefficient to calculate the heat transfer rate between fluids 1 and 2 as described in Heat Transfer Rate.

References

[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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2016a