Introduction & Context
The Overall Heat Transfer Coefficient (U) is a fundamental parameter in process engineering, representing the total resistance to heat flow between two fluids separated by a solid wall. It quantifies the efficiency of heat exchangers, boilers, and condensers. In industrial applications, calculating U is critical for sizing equipment, determining energy consumption, and ensuring process safety. This calculation accounts for convective heat transfer, fouling resistances on both surfaces, and the conductive resistance of the pipe wall material.
Methodology & Formulas
The calculation is based on the thermal resistance network method. For a cylindrical geometry, the total resistance referenced to the outer surface area (Uo) is the sum of individual resistances in series. The governing equation is derived from the reciprocal of the total thermal resistance:
\[ \frac{1}{U_o} = \frac{1}{h_o} + R_{f,o} + \left( \frac{A_o}{A_i} \right) \left( \frac{1}{h_i} + R_{f,i} \right) + \frac{A_o \ln(r_o / r_i)}{2 \pi k L} \]
Where the area ratio for a cylindrical pipe is defined by the ratio of the outer radius to the inner radius:
\[ \frac{A_o}{A_i} = \frac{r_o}{r_i} \]
The conductive resistance of the cylindrical wall, when normalized to the outer surface area, is expressed as:
\[ R_{wall} = \frac{r_o \ln(r_o / r_i)}{k} \]
The validity of the heat transfer coefficients (hi and ho) depends heavily on the flow regime, typically characterized by the Reynolds number (Re). The following table outlines the standard criteria for assessing the reliability of these calculations:
| Flow Regime |
Reynolds Number (Re) Threshold |
Reliability Status |
| Laminar |
Re < 2300 |
Low: Standard turbulent correlations may be inaccurate |
| Transition |
2300 ≤ Re < 10000 |
Moderate: Results may deviate from experimental data |
| Turbulent |
Re ≥ 10000 |
High: Calculation valid for standard correlations |
The fouling factor represents the additional thermal resistance caused by the accumulation of deposits, scale, or biological growth on heat transfer surfaces. Ignoring this factor leads to:
- An overestimation of the heat exchanger performance.
- Premature failure to meet process temperature requirements as the unit operates.
- Inadequate surface area sizing during the initial design phase.
Worked Example: Overall Heat Transfer Coefficient Calculation
A process engineer is evaluating the thermal performance of a stainless steel heat exchanger tube carrying cooling water. To determine the overall heat transfer coefficient (U) based on the outer surface area, the following parameters have been identified for the system operating at a Reynolds number of 15,000.
Knowns:
- Internal heat transfer coefficient (hi): 2000.0 W/m²K
- External heat transfer coefficient (ho): 50.0 W/m²K
- Inner diameter (di): 0.05 m
- Outer diameter (do): 0.06 m
- Wall thickness (x): 0.005 m
- Thermal conductivity of wall (k): 15.0 W/mK
- Internal fouling factor (rf,i): 0.0002 m²K/W
- External fouling factor (rf,o): 0.0001 m²K/W
Step-by-Step Calculation:
- Calculate the internal convective resistance scaled to the outer area:
\[ R_{conv,i} = \frac{1}{h_i} \times \frac{d_o}{d_i} = \frac{1}{2000} \times \frac{0.06}{0.05} = 0.0006 \text{ m²K/W} \]
- Calculate the internal fouling resistance scaled to the outer area:
\[ R_{f,i} = r_{f,i} \times \frac{d_o}{d_i} = 0.0002 \times 1.2 = 0.00024 \text{ m²K/W} \]
- Calculate the cylindrical wall resistance:
\[ R_{wall} = \frac{d_o \ln(d_o/d_i)}{2k} = \frac{0.06 \ln(0.06/0.05)}{2 \times 15} = 0.00036 \text{ m²K/W} \]
- Calculate the external convective and fouling resistances:
\[ R_{conv,o} = \frac{1}{h_o} = \frac{1}{50} = 0.02 \text{ m²K/W} \]
\[ R_{f,o} = 0.0001 \text{ m²K/W} \]
- Sum the resistances to find the total thermal resistance (Rtotal):
\[ R_{total} = 0.0006 + 0.00024 + 0.00036 + 0.02 + 0.0001 = 0.021 \text{ m²K/W} \]
- Calculate the overall heat transfer coefficient (Uo):
\[ U_o = \frac{1}{R_{total}} = \frac{1}{0.021} = 46.938 \text{ W/m²K} \]
Final Answer: The overall heat transfer coefficient (Uo) is 46.938 W/m²K.
