# Plate Heat Exchanger (TL-TL)

**Libraries:**

Simscape /
Fluids /
Heat Exchangers /
Thermal Liquid

## Description

The Plate Heat Exchanger (TL-TL) block models a plate heat exchanger between two thermal liquid networks. The block models the heat transfer between the two thermal liquids based on the effectiveness-NTU method. Heat transfer can occur in either direction between the two fluids.

The block uses the Effectiveness-NTU (E-NTU) method to model heat transfer through the
shared plate walls. You can also account for heat transfer through the plate itself by
selecting the **Plate thermal resistance** parameter. The block models
fouling on the exchanger walls, which increases thermal resistance and reduces the heat
exchange between the two fluids.

### Plate Heat Exchanger Geometry

Plate heat exchangers consist of a series of clamped-together plates. Each plate has portholes for the fluid to flow through and gaskets around the portholes and the plate sides to contain the liquid. The gasket pattern alternates between the plates so that on each side of the plate, one fluid flows across the plate while the other fluid flows through it. This results in alternating layers of hot and cold fluid. This figure shows a section of counter-flow in a plate heat exchanger.

This figure shows an individual plate and a cross-section of two adjacent plates.
In the individual plate image, the thick lines represent the gaskets and the dotted
lines represent the plate corrugation. The block assumes that the flow channel gap
between the plates equals the corrugation depth, *b*.

On this plate, one fluid flows from the top-right port hole across the plate to the bottom-right port hole. The other fluid flows through the gasket on the two left-side portholes and not touch the plate on this side. The plate corrugation patterns increase structural stiffness, surface area, and turbulence.

Because fluid flows along both sides of each plate, the total fluid volume is

$${\text{V}}_{total}=\left({N}_{p}+1\right)b{L}_{p}{W}_{p},$$

where:

*N*is the value of the_{p}**Number of plates**parameter.*L*is the value of the_{p}**Plate length**parameter.*W*is the value of the_{p}**Plate width**parameter.*b*is the value of the**Spacing between plates**parameter.

The volume of each fluid is half of the total volume. The flow cross-sectional area,
*S*, for each fluid is

$$S=\frac{\left({N}_{p}+1\right)}{2}b{W}_{p}.$$

The total projected surface area of the plates, assuming no corrugation, is

$${\text{A}}_{total,proj}=2{N}_{p}{L}_{p}{W}_{p}.$$

Each fluid has half of the total heat transfer surface area

$${A}_{T{L}_{1}}={A}_{T{L}_{2}}=\frac{{A}_{total,actual}}{2}=\varphi {N}_{p}{L}_{p}{W}_{p},$$

where the area enlargement factor, *ϕ*, accounts for the
increased surface area due to the corrugation

$$\varphi =\frac{{A}_{total,actual}}{{A}_{total,proj}}=\frac{1+\sqrt{1+{\left(\frac{\pi b}{\lambda}\right)}^{2}}+4\sqrt{1+{\left(\frac{\pi b}{\sqrt{2}\lambda}\right)}^{2}}}{6}.$$

The hydraulic diameter for each fluid is

$${\text{D}}_{h}=\frac{2b}{\varphi}\left(\frac{{N}_{p}+1}{{N}_{p}}\right).$$

### Effectiveness-NTU Heat Transfer

The heat transfer between the fluids is

$$Q=\u03f5{C}_{\text{Min}}({T}_{{\text{In,TL}}_{1}}-{T}_{{\text{In,TL}}_{2}}),$$

where:

*C*is the lesser of the heat capacity rates of the two fluids. The heat capacity rate is the product of the fluid specific heat,_{Min}*c*, and the fluid mass flow rate._{p}*C*is always positive._{Min}*T*is the zone inlet temperature of the thermal liquid 1._{In,TL1}*T*is the zone inlet temperature of the thermal liquid 2._{In,TL2}*ε*is the heat exchanger effectiveness.

Effectiveness is a function of the heat capacity rate and the number
of transfer units, *NTU*, and varies based on the heat exchanger
flow arrangement. *NTU* is

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

where:

*R*is the total thermal resistance between the two flows due to convection, conduction, and any fouling on the plates:$$R=\frac{1}{{U}_{{\text{TL}}_{1}}{A}_{{\text{TL}}_{1}}}+\frac{{F}_{{\text{TL}}_{1}}}{{A}_{{\text{TL}}_{1}}}+{R}_{\text{W}}+\frac{{F}_{{\text{TL}}_{2}}}{{A}_{{\text{TL}}_{2}}}+\frac{1}{{U}_{{\text{TL}}_{2}}{A}_{{\text{TL}}_{2}}},$$

where:

*U*is the convective heat transfer coefficient of the respective thermal liquid.*F*is the value of the**Fouling factor**parameter on the thermal liquid 1 or 2 side.*R*is the thermal resistance through the plates. If you select_{W}**Plate thermal resistance**, $${\text{R}}_{W}=\frac{t}{{N}_{p}{L}_{p}{W}_{p}{k}_{p}},$$ where*t*is the value of the**Plate thickness**parameter and*k*is the value of the_{p}**Plate thermal conductivity**parameter.If you clear the

**Plate thermal resistance**check box,*R*._{W}= 0

**Effectiveness by Flow Arrangement**

The heat exchanger effectiveness varies according to the flow configuration. The effectiveness is $$\epsilon =1-\mathrm{exp}(-NTU)$$ for all flow arrangements.

When **Flow arrangement** is `Parallel flow`

$$\u03f5=\frac{1-\text{exp}[-NTU(1+{C}_{\text{R}})]}{1+{C}_{\text{R}}}.$$

When **Flow arrangement** is `Counter flow`

$$\u03f5=\frac{1-\text{exp}[-NTU(1-{C}_{\text{R}})]}{1-{C}_{\text{R}}\text{exp}[-NTU(1-{C}_{\text{R}})]}.$$

*C*_{R} is the ratio between the heat
capacity rates of the two fluids:

$${C}_{\text{R}}=\frac{{C}_{\text{Min}}}{{C}_{\text{Max}}}.$$

### Pressure Loss

For both fluids and pressure loss parameterizations, the pressure loss between each port and the internal node is

$$\begin{array}{l}{\text{p}}_{A}-{p}_{I}=\frac{{\xi}_{ports}{\dot{m}}_{A}\left|{\dot{m}}_{A}\right|}{4\rho {S}_{port,A}^{2}}+\frac{{f}_{Darcy}{L}_{p}{\dot{m}}_{A}\left|{\dot{m}}_{A}\right|}{4\rho {D}_{h}{S}^{2}}\\ {\text{p}}_{B}-{p}_{I}=\frac{{\xi}_{ports}{\dot{m}}_{B}\left|{\dot{m}}_{B}\right|}{4\rho {S}_{port,B}^{2}}+\frac{{f}_{Darcy}{L}_{p}{\dot{m}}_{B}\left|{\dot{m}}_{B}\right|}{4\rho {D}_{h}{S}^{2}}\end{array}$$

where:

*ṁ*is the mass flow rate through port_{A}**A1**or**A2**when the block calculates the values for the thermal liquid 1 and thermal liquid 2 networks, respectively.*ṁ*is the mass flow rate through port_{B}**B1**or**B2**, when the block calculates the values for the thermal liquid 1 and thermal liquid 2 networks, respectively.*ρ*is the fluid density.*ξ*is the value of the_{ports}**Loss coefficient for inlet and outlet manifolds and ports**parameter.*S*is the value of the_{port,A}**Cross-sectional area at port A1**or**Cross-sectional area at port A2**parameter, when the block calculates the values for the thermal liquid 1 and thermal liquid 2 networks, respectively.*S*is the value of the_{port,B}**Cross-sectional area at port B1**or**Cross-sectional area at port B2**parameter, when the block calculates the values for the thermal liquid 1 and thermal liquid 2 networks, respectively.

When the **Pressure loss model** parameter is ```
Friction
factor
```

, you use the **Darcy friction factor for flow
between corrugated plates** parameter to specify a single constant
friction factor for the fluid. *f _{Darcy}* is
the value of the

**Darcy friction factor for flow between corrugated plates**parameter.

When the **Pressure loss model** parameter is
`Correlation for flow between corrugated plates`

, the
block calculates the Darcy friction factor based on the plate geometry data and
Reynolds number from [3].

The Reynolds number is $$\text{Re=}\frac{\dot{m}{D}_{h}}{\mu S},$$ where *μ* is the dynamic viscosity. When the
Reynolds number is between 200 and 1000,

$${f}_{Darcy}={\left[\frac{\text{cos}\beta}{\sqrt{0.18\mathrm{tan}\beta +0.36\mathrm{sin}\beta +\frac{64}{Re\text{cos}\beta}}}+\frac{1-\text{cos}\beta}{\sqrt{3.8\left(\frac{597}{Re}+3.85\right)}}\right]}^{-2}.$$

When the Reynolds number is greater than 2000,

$${f}_{Darcy}={\left[\frac{\text{cos}\beta}{\sqrt{0.18\mathrm{tan}\beta +0.36\mathrm{sin}\beta +\frac{1}{{\left(1.8\mathrm{ln}Re-1.5\right)}^{2}\text{cos}\beta}}}+\frac{1-\text{cos}\beta}{\sqrt{\frac{3.8\ast 39}{{\mathrm{Re}}^{0.289}}}}\right]}^{-2}.$$

The block uses a smooth cubic transition between these regions and for Reynolds numbers
between 1000 and 2000. For Reynolds numbers below 200, the block calculates
*f _{Darcy}* from a laminar friction
constant that ensures numerical robustness as the flow goes to 0.

### Heat Transfer Coefficient

The heat transfer coefficient, *U*, is

$$U=Nu\frac{k}{{D}_{h}},$$

where *k* is the fluid thermal conductivity. The
heat transfer calculations depend on the value of the **Heat transfer
coefficient model** parameter.

**Colburn Equation**

When **Heat transfer coefficient model** is
`Colburn equation`

, the block calculates the heat
transfer coefficient from data by using the Colburn correlation
coefficients.

The Nusselt number is

$$Nu=aR{e}^{b}P{r}^{c},$$

where *a*, *b*, and
*c* are the first, second, and third elements in the
**Coefficients [a, b, c] for a*Re^b*Pr^c** parameter and
*Pr* is the Prandtl number.

**Correlation for Flow Between Corrugated Plates**

When **Heat transfer coefficient model** is
`Correlation for flow between corrugated plates`

,
the block calculates the heat transfer coefficient based on the plate geometry
data. The Nusselt number from [3] is

$$Nu={c}_{1}{\left({f}_{Darcy}R{e}^{2}\mathrm{sin}2\beta \right)}^{{c}_{2}}{\mathrm{Pr}}^{{c}_{3}},$$

where:

*c*,_{1}*c*, and_{2}*c*are the first, second, and third elements in the_{3}**Coefficients for Martin correlation**parameter.*Pr*is the Prandtl number.

### Conservation Equations

The total mass accumulation rate for both thermal liquids is

$$\begin{array}{l}\frac{d{M}_{{\text{TL}}_{1}}}{dt}={\dot{m}}_{\text{A1}}+{\dot{m}}_{\text{B1}}\\ \frac{d{M}_{{\text{TL}}_{2}}}{dt}={\dot{m}}_{\text{A2}}+{\dot{m}}_{\text{B2}}\end{array}$$

where $$\dot{m}$$_{i} is the mass flow rate of the fluid through
a port **i**.

The energy conservation equations are

$$\begin{array}{l}{M}_{T{L}_{1}}\frac{d{u}_{T{L}_{1}}}{dt}+{u}_{T{L}_{1}}\left({\dot{m}}_{A1}+{\dot{m}}_{B1}\right)={\varphi}_{\text{A1}}+{\varphi}_{\text{B1}}-Q\\ {M}_{T{L}_{2}}\frac{d{u}_{T{L}_{2}}}{dt}+{u}_{T{L}_{2}}\left({\dot{m}}_{A2}+{\dot{m}}_{B2}\right)={\varphi}_{\text{A2}}+{\varphi}_{\text{B2}}+Q\end{array}$$

where *ϕ _{i}* is the energy
flow rate at a port

**i**.

The heat transferred to or from one thermal liquid, *Q*, equals
the heat transferred from or to the other thermal liquid.

## Ports

### Conserving

## Parameters

## References

[1] Shah, Ramesh K., and Dusan P.
Sekulic. *Fundamentals of heat exchanger design*. John Wiley &
Sons, 2003.

[2] Longo, Giovanni A., Giulia
Righetti, and Claudio Zilio. "A new computational procedure for refrigerant condensation
inside herringbone-type Brazed Plate Heat Exchangers." *International Journal
of Heat and Mass Transfer* 82 (2015): 530-536.

[3] Martin, Holger. "A theoretical
approach to predict the performance of chevron-type plate heat exchangers."
*Chemical Engineering and Processing: Process Intensification*
35.4 (1996): 301-310.

## Extended Capabilities

## Version History

**Introduced in R2024a**