Introduction & Context
Heat exchanger duty calculation is a fundamental task in process engineering, serving as the primary method for determining the thermal energy transfer rate required to achieve a specific temperature change in a process fluid. This calculation is critical for the sizing, selection, and performance evaluation of shell-and-tube heat exchangers, plate heat exchangers, and other thermal management equipment. By establishing the heat duty (Q), engineers can determine the necessary surface area (A) and the overall heat transfer coefficient (U) required to maintain process efficiency, ensure equipment safety, and optimize energy consumption in industrial systems such as chemical reactors, power plants, and HVAC installations.
Methodology & Formulas
The calculation follows a systematic approach based on the first law of thermodynamics and the thermal resistance network model.
First, the heat duty is determined by the energy balance of the fluid:
\[ Q = \dot{m} \cdot c_p \cdot (T_{out} - T_{in}) \]
The overall heat transfer coefficient (U) is derived by summing the individual thermal resistances, including internal convection, wall conduction, and external convection, normalized to the inner surface area:
\[ \frac{1}{U} = \frac{1}{h_i} + \frac{r_i \cdot \ln(r_o / r_i)}{k_{tube}} + \left( \frac{r_i}{r_o} \right) \cdot \frac{1}{h_o} \]
Finally, the required surface area is calculated using the Log Mean Temperature Difference (LMTD) method:
\[ A = \frac{Q}{U \cdot \Delta T_{lm}} \]
| Parameter |
Condition/Threshold |
Engineering Significance |
| Reynolds Number (Re) |
Re < 10,000 |
Turbulent flow correlations may be invalid; transition or laminar flow regime. |
| Temperature Gradient |
Tout ≤ Tin |
Invalid for heating duty; indicates no heat gain or potential cooling process. |
| Resistance Summation |
Denominator > 0 |
Ensures numerical stability; prevents division by zero in U calculation. |
In an ideal steady-state scenario, the heat lost by the hot fluid must equal the heat gained by the cold fluid. If your calculations show a discrepancy, investigate the following:
- Inaccurate physical property data for the fluids at the operating temperature.
- Measurement errors in flow rates or temperature sensors.
- Unaccounted phase changes or chemical reactions occurring within the exchanger.
- Significant heat loss to the surroundings or through support structures.
Worked Example: Heat Exchanger Duty and Sizing
A process engineer is tasked with evaluating the performance of a single-pass shell-and-tube heat exchanger used to heat process water. The system must raise the temperature of a water stream from 20.000 °C to 72.000 °C at a mass flow rate of 0.500 kg/s. The following parameters define the system configuration:
Knowns:
- Mass flow rate (m_dot): 0.500 kg/s
- Specific heat capacity of water (cp_water): 4180.000 J/(kg·K)
- Inlet temperature (t_in): 20.000 °C
- Outlet temperature (t_out): 72.000 °C
- Log mean temperature difference (delta_t_lm): 15.000 °C
- Inner heat transfer coefficient (h_i): 2500.000 W/(m²·K)
- Outer heat transfer coefficient (h_o): 1200.000 W/(m²·K)
- Inner tube radius (r_i): 0.010 m
- Outer tube radius (r_o): 0.012 m
- Thermal conductivity of tube material (k_tube): 16.000 W/(m·K)
Step-by-Step Calculation:
-
Calculate the heat duty (Q) required for the process:
\[ Q = \dot{m} \cdot c_p \cdot (T_{out} - T_{in}) \]
\[ Q = 0.500 \cdot 4180.000 \cdot (72.000 - 20.000) = 108680.000 \text{ W} \]
-
Determine the individual thermal resistances:
- Inner convection: \( R_i = 1 / h_i = 0.0004 \text{ m²·K/W} \)
- Wall conduction: \( R_w = (r_o - r_i) / k_{tube} = (0.012 - 0.010) / 16.000 = 0.000114 \text{ m²·K/W} \)
- Outer convection: \( R_o = 1 / h_o = 0.000694 \text{ m²·K/W} \)
-
Calculate the overall heat transfer coefficient (U):
\[ U = 1 / (R_i + R_w + R_o) \]
\[ U = 1 / (0.0004 + 0.000114 + 0.000694) = 827.544 \text{ W/(m²·K)} \]
-
Calculate the required heat transfer area (A):
\[ A = Q / (U \cdot \Delta T_{lm}) \]
\[ A = 108680.000 / (827.544 \cdot 15.000) = 8.755 \text{ m²} \]
Final Answer:
The required heat duty for the exchanger is 108680.000 W, necessitating a total heat transfer area of 8.755 m² to satisfy the process requirements.
