# Effectiveness-NTU Calculator

Created by Gabriela Diaz
Reviewed by Anna Szczepanek, PhD and Jack Bowater
Last updated: Jan 18, 2024

With Omni's effectiveness-NTU calculator, you'll be able to execute design and performance calculations of heat exchangers. With the design calculations, you can determine the heat transfer area (A) for a given set of conditions. Use the performance calculation to find the outlet temperatures (Tco and Tho) of the fluids and the actual heat transfer rate (q).

If you'd like to learn a bit more about this method, we invite you to keep reading and find out about:

• What is the effectiveness-NTU method;
• Its advantages over the LMTD method;
• How to calculate the effectiveness of a heat exchanger;
• The formula for the effectiveness of a heat exchanger;
• How to calculate the NTU of a heat exchanger; and
• The NTU and ε formulas for different heat exchangers configurations.

## What is the effectiveness-NTU method?

Similar to the LMTD (log mean temperature difference), the effectiveness-NTU method is a method used to analyze heat exchangers. This one is preferred when the outlet temperatures of the fluids are unknown, since, in these cases, the LMTD requires a cumbersome iterative solution.

💡 You can read more about the LMTD method and how to calculate the LMTD for different heat exchanger configurations with the LMTD calculator.

The term effectiveness (ε) is a dimensionless indicator that relates the actual heat transfer rate (q) to the maximum possible heat transfer rate (qmax) that could occur for a particular heat exchanger and a particular set of fluids. The formula for the effectiveness of a heat exchanger is given by the ratio of these heats:

$\footnotesize \varepsilon = \dfrac {q}{q_{max}}$

The actual heat rate $q$ is determined in terms of the associated properties as:

\begin{align*} \footnotesize q = C_{c} \cdot (T_{co} - T_{ci}) \\ \footnotesize = C_{h} \cdot (T_{hi} - T_{ho}) \end{align*}

Where:

• $T_{ci}$ and $T_{co}$ - Inlet and outlet temperatures of the cold fluid, respectively;
• $C_c$ - Heat capacity of the cold fluid;
• $T_{hi}$ and $T_{ho}$ - Inlet and outlet temperatures of the hot fluid, respectively; and
• $C_h$ - Heat capacity of the hot fluid.

If you're unsure about what heat capacity is, you can always check the heat capacity calculator and learn more about this subject.

In order to calculate the maximum possible heat transfer rate (qmax) that could happen in a given heat exchanger, the following expression is typically used:

$\footnotesize q_{max} = C_{min} \cdot (T_{hi} - T_{ci})$

Where $C_{min}$ corresponds to the minimum value of the heat capacities, i.e., whichever is smaller of $C_c$ or $C_h$.

This heat rate is calculated using the difference between the inlet temperature of the hot fluid, $T_{hi}$, and the inlet temperature of the cold fluid, $T_{ci}$. This represents the maximum temperature difference in the system.

Why is $q_{max}$ calculated with $C_{min}$ and not with $C_{max}$? 🤔 The reason for this is that the fluid with the lowest heat capacity is the one that can experience the maximum temperature change. This could mean receiving or giving heat, whether it's the cold or the hot fluid.

The number of transfer units (NTU) is another dimensionless parameter used in the effectiveness-NTU method. It is defined in terms of the overall heat transfer coefficient, $U$, the area of heat transfer, $A$, and the minimum heat capacity, $C_{min}$:

$\footnotesize NTU = \dfrac {U\cdot A}{C_{min}}$

Finally, another dimensionless parameter is used in the effectiveness-NTU method. This one is known as the ratio of the heat capacities, (Cr), which is given by:

$\footnotesize C_r = \dfrac {C_{min}}{C_{max}}$

## How to use the effectiveness-NTU calculator

With the effectiveness-NTU calculator, you'll be able to perform either design calculations or performance assessments of a specific heat exchanger.

The objective of a design calculation is to find the total area of heat transfer (A) required for a given set of inlet and outlet temperatures. With the performance calculation, you'll be able to find the actual heat transfer rate (q) and the outlet temperatures of both the cold (Tco) and hot (Tho) fluids.

With these distinctions in mind, let's see how to use the effectiveness-NTU calculator for both cases.

### Design calculation - determine area (A)

To perform a design calculation, follow these steps:

1. Select the Design problem option from the Type of calculation field.
2. On the row below, choose the type of heat exchanger that you'd like to evaluate.
3. Next, you'll find two blocks, Cold fluid properties and Hot fluid properties, where you'll be able to input the respective properties of the cold and hot fluids. Here, enter the known properties of the fluids, such as mass, inlet and outlet temperatures, and specific heat.
4. At this point, the calculator will be able to show you the results for the actual heat transfer rate (q), the maximum heat transfer rate (qmax), and the effectiveness (ε) of the chosen device.
5. Finally, in the Area of heat transfer block, enter the value of the overall heat coefficient (U). Once you've input it, the calculator will be able to determine the NTU value and compute the area of heat transfer (A).
6. That's it! 😀

### Performance assessment - calculate heat transfer rate (q) and outlet temperatures (Tco and Tho)

On the other hand, if you'd like to execute the performance calculations, this is what you have to do:

1. On the Type of calculation row, select the Performance calculation option.
2. On the field below, choose the type of heat exchanger that you'd like to assess.
3. Next, on the cold and hot fluids properties' blocks Cold fluid properties and Hot fluid properties, enter the respective known properties of the substances: mass, inlet temperature, and specific heat.
4. Next, on the Area of heat transfer block, input the values for the overall heat coefficient (U) and area of heat transfer (A).
5. After all these values have been entered, the calculator will show you the results for the outlet temperatures (Tco and Tho) and the actual heat transfer rate (q).

🙋 The effectiveness-NTU calculator also shows the result of other in-between steps, such as the value for the effectiveness (ε), the NTU, and the ratio of the heat capacities (Cr). This way, you can check your results if you're doing the calculations yourself 😉

## How do I determine the effectiveness of a heat exchanger?

To determine the effectiveness of a heat exchanger, there're two possibilities:

Design calculations

1. From fluid's properties, calculate the maximum (qmax) and actual heat transfer (q).
2. Determine the ratio between the heat capacities of the fluids, Cr = Cmin / Cmáx.
3. Calculate the effectiveness as the ratio of the heats, ε = q/qmax.

Performance calculations

1. Determine Cr as: Cr = Cmin / Cmáx.
2. Calculate NTU using the relationship: NTU = U x A / Cmin
3. With the values of NTU and Cmin, use the ε formula or curves of the respective heat exchanger to calculate ε.

💡 If you don't know the value of the overall heat transfer coefficient (U), you can use our heat transfer coefficient calculator to determine it.

## How do I calculate the NTU of a heat exchanger?

To calculate the NTU of a heat exchanger, there're two possible situations:

Design calculations

1. From the fluid's properties, calculate the maximum (qmax) and actual heat transfer (q) rates.
2. Determine the ratio between the heat capacities of the fluids, Cr = Cmin / Cmáx.
3. Calculate the effectiveness as a ratio of the heats, ε = q/qmax.
4. With the values of ε and Cr, use the NTU formula or curves of the respective heat exchanger to calculate NTU.

Performance calculations

1. Determine Cmin from the fluids' properties.
2. Calculate NTU with the relationship: NTU = U x A / Cmin

As you've might have noticed, depending upon the type of calculations that you're interested in performing, the formulas to determine the parameters of effectiveness (ε) and number of transfer units (NTU) are not always the short versions that we indicated earlier in the definitions section of this article. Instead, these are calculated from formulas or charts.

In the following sections, you can find the expressions for the most commonly used configurations of heat exchangers.

## NTU formulas for different types of heat exchangers

To determine the number of transfer units (NTU), you could either use a graph or a formula for a specific type of heat exchanger. In the table below, you can find the expressions to calculate NTU:

Type of heat exchanger

Formula

Parallel flow

$\footnotesize NTU = \dfrac{-\ln(1- \varepsilon\cdot(1 + C_r))}{1 + C_r}$

Counter flow

If $\footnotesize C_r <1$, use:

$\footnotesize NTU = \dfrac{1}{C_r -1} \cdot \ln\bigg (\dfrac{\varepsilon - 1}{\varepsilon C_r - 1}\bigg)$

If $\footnotesize C_r = 1$, use:

$\footnotesize NTU = \dfrac{\varepsilon}{1- \varepsilon}$

Shell and tube (One shell pass and 2, 4, ... tube passes)

$\footnotesize A = 1 + {C_r}^2$

$\footnotesize NTU_1 = -A^{-\frac 12} \cdot \ln \bigg(\dfrac{E - 1}{E+1}\bigg)$

$\footnotesize E = \dfrac{2/\varepsilon_1 - (1 + C_r)}{(1 + {C_r}^2)^{\frac 12}}$

Shell and tube (n shell passes and 2n, 4n, ... tube passes)

$\footnotesize \varepsilon_1 = \dfrac{F-1}{F-C_r}$

$F ={\bigg( \dfrac{\varepsilon C_r-1}{\varepsilon - 1}\bigg)}^{\frac 1n}$

$\footnotesize NTU = n\cdot NTU_1$

Cross flow (Cmax (mixed), Cmin (unmixed))

$\footnotesize B = \ln(1- \varepsilon C_r)$

$\footnotesize NTU = - \ln\bigg(1+\dfrac{B}{C_r}\bigg)$

Cross flow (Cmax (unmixed), Cmin (mixed))

$\footnotesize D = \ln(1- \varepsilon)$

$\footnotesize NTU = \dfrac{-1}{C_r}\ln[C_r D + 1]$

All exchangers (Cr = 0)

$\footnotesize NTU = - \ln(1 - \varepsilon)$

## Effectiveness formulas for different types of heat exchangers

Similar to the NTU, in order to calculate the effectiveness (ε), you can use a graph or a formula for a given type of heat exchanger. In the table below, you'll find the expressions to calculate ε:

Type of heat exchanger

Formula

Parallel flow

$\footnotesize \varepsilon = \dfrac{1-\exp[-NTU(1+C_r)]}{1+C_r}$

Counter flow

If $\footnotesize C_r <1$, use:

$\footnotesize G = (1-C_r)$

$\footnotesize \varepsilon = \dfrac{1-\exp[-NTU\cdot G]}{1-C_r \cdot \exp[-NTU\cdot G]}$

If $\footnotesize C_r = 1$, use:

$\footnotesize \varepsilon = \dfrac{NTU}{1+NTU}$

Shell and tube (One shell pass and 2, 4, ... tube passes)

$\footnotesize A = 1 +{C_r}^2$

$\footnotesize H = 1 +C_r$

$\footnotesize J = \dfrac{1-\exp[-NTU_1\cdot({A}^{\frac 12})]}{1 +\exp[-NTU_1\cdot({A}^{\frac 12})]}$

$\footnotesize \varepsilon_1 = 2\cdot(H + A \cdot J)$

Shell and tube (n shell passes and 2n, 4n, ... tube passes)

$\footnotesize K = {\bigg(\dfrac{1-\varepsilon_1 C_r}{1-\varepsilon_1}\bigg)}^{n}$

$\footnotesize \varepsilon = \dfrac{K-1}{K-C_r}$

Cross flow (both fluids unmixed)

$\footnotesize L = NTU^{0.22}[\exp(-C_r\cdot {NTU}^{0.78})$

$\footnotesize \varepsilon = 1 - \exp\bigg[ \dfrac{L}{C_r} - 1\bigg]$

Cross flow (Cmax (mixed), Cmin (unmixed))

$\footnotesize M = \exp[-C_r(1-\exp(-NTU))]$

$\footnotesize \varepsilon = \dfrac{1-M}{C_r}$

Cross flow (Cmax (unmixed), Cmin (mixed))

$\footnotesize N = 1-\exp(-C_r \cdot NTU)$

$\footnotesize \varepsilon = 1-\exp \bigg(\dfrac{-N}{C_r}\bigg)$

All exchangers (Cr = 0)

$\footnotesize \varepsilon = 1-\exp(-NTU)$

## FAQ

### What are the advantages of the NTU method over the LMTD method?

The main advantage of the NTU method over the LMTD method is that for performance calculations, i.e., determining heat transfer rate and outlet temperatures, the LMTD requires an iterative solution, while with the NTU, the solution can be obtained directly from the formulas.

### Can the effectiveness of a heat exchanger be greater than 1?

No, the effectiveness of a heat exchanger can't be greater than 1. The effectiveness (ε) of a heat exchanger should always be a value between zero and one, 0 < ε < 1. The effectiveness represents the ratio between the actual heat rate (q) and the maximum possible heat transfer rate (qmax) that can occur in a heat exchanger for a given set of fluids' conditions (ε = q/qmax).

Gabriela Diaz
Type of calculation
Design problem
Heat exchanger
Parallel flow
Cold fluid properties
Mass flow rate (m꜀)
lb/s
Specific heat capacity (cₚ꜀)
BTU/(lb·°F)
Heat capacity (C꜀)
BTU/°F
Inlet temperature (T꜀ᵢ)
°F
Outlet temperature (T꜀ₒ)
°F
Hot fluid properties
Mass flow rate (mₕ)
lb/s
Specific heat capacity (cₚₕ)
BTU/(lb·°F)
Heat capacity (Cₕ)
BTU/°F
Inlet temperature (Tₕᵢ)
°F
Outlet temperature (Tₕₒ)
°F
Heat transfer summary
Actual heat rate (q)
BTU/h
Area of heat transfer
Overall heat transfer coeff. (U)
BTU/(h·ft²·°F)
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