# Plate Heat Exchanger (TL-TL)

Plate geometry heat exchanger between two thermal liquid networks

Since R2024a

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:

• Np is the value of the Number of plates parameter.

• Lp is the value of the Plate length parameter.

• Wp is the value of the 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

`$Q=ϵ{C}_{\text{Min}}\left({T}_{{\text{In,TL}}_{1}}-{T}_{{\text{In,TL}}_{2}}\right),$`

where:

• CMin is the lesser of the heat capacity rates of the two fluids. The heat capacity rate is the product of the fluid specific heat, cp, and the fluid mass flow rate. CMin is always positive.

• TIn,TL1 is the zone inlet temperature of the thermal liquid 1.

• TIn,TL2 is the zone inlet temperature of the thermal liquid 2.

• 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.

• RW is the thermal resistance through the plates. If you select 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 kp is the value of the Plate thermal conductivity parameter.

If you clear the Plate thermal resistance check box, RW = 0.

Effectiveness by Flow Arrangement

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

When Flow arrangement is `Parallel flow`

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

When Flow arrangement is `Counter flow`

`$ϵ=\frac{1-\text{exp}\left[-NTU\left(1-{C}_{\text{R}}\right)\right]}{1-{C}_{\text{R}}\text{exp}\left[-NTU\left(1-{C}_{\text{R}}\right)\right]}.$`

CR 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

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

where:

• A is the mass flow rate through port A1 or A2 when the block calculates the values for the thermal liquid 1 and thermal liquid 2 networks, respectively.

• B is the mass flow rate through port 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.

• ξports is the value of the Loss coefficient for inlet and outlet manifolds and ports parameter.

• Sport,A is the value of the 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.

• Sport,B is the value of the 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. fDarcy 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 $\text{Re=}\frac{\stackrel{˙}{m}{D}_{h}}{\mu S},$ 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 fDarcy 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:

• c1, c2, and c3 are the first, second, and third elements in the Coefficients for Martin correlation parameter.

• Pr is the Prandtl number.

### Conservation Equations

The total mass accumulation rate for both thermal liquids is

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

where $\stackrel{˙}{m}$i is the mass flow rate of the fluid through a port i.

The energy conservation equations are

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

where ϕi 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

expand all

Inlet or outlet port associated with thermal liquid 1.

Inlet or outlet port associated with thermal liquid 1.

Inlet or outlet port associated with thermal liquid 2.

Inlet or outlet port associated with thermal liquid 2.

## Parameters

expand all

### Configuration

Flow path alignment across the heat exchanger. The available flow arrangements are:

• `Parallel flow` — The flows run in the same direction.

• `Counter flow` — The flows run parallel to each other, in the opposite directions.

The e-NTU calculations are based on the value of this parameter, regardless of the actual flow direction between ports A1 and B1 and between ports A2 and B2 during simulation.

Number of plates in the heat exchanger. Fluid flows across both sides of each plate, including the end plate and start plate.

Length of each plate. This value is the longer side of the plate and the flow axis of the fluid.

Width of each plate.

Gap between each plate. This parameter value usually equals the corrugation depth.

Whether you specify the surface area enlargement factor directly or the block calculates the enlargement factor from a specified chevron geometry.

Multiplier that accounts for the increase in plate surface area due to corrugation. If each plate is perfectly flat, the value of this parameter is 1. Otherwise, this value is the ratio of the actual total surface area to the area of the two-dimensional projection of the plate shape.

#### Dependencies

To enable this parameter, set Corrugation pattern specification to ```Surface area enlargement factor```.

Angle of the chevron corrugation on each plate. This value is β in this plate diagram:

#### Dependencies

To enable this parameter, set Corrugation pattern specification to ```Chevron geometry```.

Ratio of the chevron corrugation depth to the corrugation pitch. The corrugation pitch is the corrugation wavelength. The value of this parameter is b/λ in this diagram:

#### Dependencies

To enable this parameter, set Corrugation pattern specification to ```Chevron geometry```.

Option to model plate thermal resistance. If you select this check box, the block uses the values of the Plate thickness and Plate thermal conductivity parameters to calculate the thermal resistance through each plate. If you clear this check box, the block does not model the effects of thermal resistance through the plates.

Thickness of each plate in the heat exchanger.

#### Dependencies

To enable this parameter, select Plate thermal resistance.

Thermal conductivity of each plate.

#### Dependencies

To enable this parameter, select Plate thermal resistance.

Flow area at the thermal liquid 1 port A1.

Flow area at the thermal liquid 1 port B1.

Flow area at the thermal liquid 2 port A2.

Flow area at the thermal liquid 2 port B2.

### Thermal Liquid 1

Method of pressure loss calculation due to viscous friction for thermal liquid 1. The settings are:

• `Friction factor` — The block calculates the pressure loss based on a constant Darcy friction factor for flow between the plate passages, specified by the Darcy friction factor for flow between corrugated plates parameter.

• ```Correlation for flow between corrugated plates``` — The block uses the Martin correlation for plate heat exchangers with the chevron or herringbone pattern to calculate the pressure loss.

Constant Darcy friction factor to use to calculate the pressure loss.

If you have data for the pressure loss across the heat exchanger, Δpdata, and the mass flow rate through the heat exchanger, data, the value of this parameter is

`${f}_{darcy}=\frac{{D}_{h}{S}^{2}}{{L}_{p}}\left[2\rho \left(\frac{\Delta {p}_{data}}{{\stackrel{˙}{m}}_{data}^{2}}\right)-\frac{{\xi }_{ports}}{{S}_{port}}\right].$`

If you do not have data for ξports, use the default value of the Loss coefficient for inlet and outlet manifolds and ports parameter in this expression. If you have multiple data points for Δpdata and data, use a least squares fit to find fdarcy.

If you know the loss coefficient value, ξloss, for your heat exchanger, then fDarcy = ξloss Dh / LP.

#### Dependencies

To enable this parameter, set Pressure loss model to ```Friction factor```.

Pressure loss coefficient that accounts for losses at the inlet and outlet manifolds and ports on the thermal liquid 1 side.

Method of calculating the heat transfer coefficient for thermal liquid 1. The available settings are:

• `Colburn equation` — Use this setting to calculate the heat transfer coefficient by defining the variables a, b, and c of the Colburn equation. Use this option to fit the heat transfer results to the experimental data by estimating the coefficients.

• ```Correlation for flow between corrugated plates``` — Use this setting to calculate the heat transfer coefficient based on the plate geometry data and the Martin correlation.

Three-element vector that contains the empirical coefficients of the Colburn equation for the thermal liquid 1. The Colburn equation is a formulation for calculating the Nusselt number.

#### Dependencies

To enable this parameter, set Heat transfer coefficient model to ```Colburn equation```.

Three-element vector that contains the coefficients c1, c2, and c3 that the block uses to calculate the Nusselt number by using the Martin correlation for the thermal liquid 1:

`${\text{Nu=c}}_{1}{\left({f}_{Darcy}{\mathrm{Re}}^{2}\mathrm{sin}2\beta \right)}^{{c}_{2}}{\mathrm{Pr}}^{{c}_{3}}.$`

#### Dependencies

To enable this parameter, set Heat transfer coefficient model to ```Correlation for flow between corrugated plates```.

Additional thermal resistance due to fouling layers on the surfaces of the wall for the thermal liquid 1. In real systems, fouling deposits grow over time. However, the growth is slow enough that the block assumes the fouling is constant during the simulation.

Thermal liquid 1 pressure at the start of the simulation.

Temperature in the thermal liquid 1 at the start of the simulation. This parameter can be a scalar or a two-element vector. A scalar value represents the mean initial temperature in the fluid flow. A vector value represents the initial temperature at the inlet and outlet in the form [`inlet`, `outlet`]. The block calculates a linear gradient between the two ports. The initial flow direction determines the inlet and outlet ports.

### Thermal Liquid 2

Method of pressure loss calculation due to viscous friction for thermal liquid 2. The settings are:

• `Friction factor` — The block calculates the pressure loss based on a constant Darcy friction factor for flow between the plate passages, specified by the Darcy friction factor for flow between corrugated plates parameter.

• ```Correlation for flow between corrugated plates``` — The block uses the Martin correlation for plate heat exchangers with the chevron or herringbone pattern to calculate the pressure loss.

Constant Darcy friction factor to use to calculate the pressure loss.

If you have data for the pressure loss across the heat exchanger, Δpdata, and the mass flow rate through the heat exchanger, data, the value of this parameter is

`${f}_{darcy}=\frac{{D}_{h}{S}^{2}}{{L}_{p}}\left[2\rho \left(\frac{\Delta {p}_{data}}{{\stackrel{˙}{m}}_{data}^{2}}\right)-\frac{{\xi }_{ports}}{{S}_{port}}\right].$`

If you do not have data for ξports, use the default value of the Loss coefficient for inlet and outlet manifolds and ports parameter in this expression. If you have multiple data points for Δpdata and data, use a least squares fit to find fdarcy.

If you know the loss coefficient value, ξloss, for your heat exchanger, then fDarcy = ξloss Dh / LP.

#### Dependencies

To enable this parameter, set Pressure loss model to ```Friction factor```.

Pressure loss coefficient that accounts for losses at the inlet and outlet manifolds and ports on the thermal liquid 2 side.

Method of calculating the heat transfer coefficient for thermal liquid 2. The available settings are:

• `Colburn equation` — Use this setting to calculate the heat transfer coefficient by defining the variables a, b, and c of the Colburn equation. Use this option to fit the heat transfer results to the experimental data by estimating the coefficients.

• ```Correlation for flow between corrugated plates``` — Use this setting to calculate the heat transfer coefficient based on the plate geometry data and the Martin correlation.

Three-element vector that contains the empirical coefficients of the Colburn equation for the thermal liquid 2. The Colburn equation is a formulation for calculating the Nusselt number.

#### Dependencies

To enable this parameter, set Heat transfer coefficient model to ```Colburn equation```.

Three-element vector that contains the coefficients c1, c2, and c3 that the block uses to calculate the Nusselt number by using the Martin correlation for the thermal liquid 2:

`${\text{Nu=c}}_{1}{\left({f}_{Darcy}{\mathrm{Re}}^{2}\mathrm{sin}2\beta \right)}^{{c}_{2}}{\mathrm{Pr}}^{{c}_{3}}.$`

#### Dependencies

To enable this parameter, set Heat transfer coefficient model to ```Correlation for flow between corrugated plates```.

Additional thermal resistance due to fouling layers on the surfaces of the wall for the thermal liquid 2. In real systems, fouling deposits grow over time. However, the growth is slow enough that the block assumes the fouling is constant during the simulation.

Thermal liquid 2 pressure at the start of the simulation.

Temperature in the thermal liquid 1 at the start of the simulation. This parameter can be a scalar or a two-element vector. A scalar value represents the mean initial temperature in the fluid flow. A vector value represents the initial temperature at the inlet and outlet in the form [`inlet`, `outlet`]. The block calculates a linear gradient between the two ports. The initial flow direction determines the inlet and outlet ports.

## 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.

## Version History

Introduced in R2024a