Overall Heat Transfer Coefficient (U) Estimation

Reference ID: MET-1047 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The Overall Heat Transfer Coefficient (U) is a critical parameter in Process Engineering, representing the total thermal resistance of a multi‑layered system. In the context of a steam retort, this calculation quantifies the rate of heat transfer from the condensing steam environment to the food product inside a sealed container. Accurate estimation of U is essential for determining thermal processing times, ensuring food safety (sterilization), and optimizing energy efficiency in batch processing operations. Moreover, complying with thermal process steam quality requirements is crucial to maintain consistent heat transfer performance.

Methodology & Formulas

The calculation utilizes a resistance network model, where the total thermal resistance is the sum of individual resistances in series. The governing equation for the fouled overall heat transfer coefficient is defined as:

\[ \frac{1}{U_{fouled}} = \frac{1}{h_{steam}} + \frac{x_{wall}}{k_{wall}} + R''_{fouling} + \frac{1}{h_{food}} \]

The individual components are derived as follows:

Steam-side film coefficient (\(h_{steam}\)): Calculated using Nusselt’s correlation for film condensation on a vertical surface:
\[ h_{steam} = 0.943 \cdot \left[ \frac{k_{liq}^3 \cdot \rho_{liq} \cdot (\rho_{liq} - \rho_{vap}) \cdot g \cdot h'_{fg}}{\mu_{liq} \cdot H \cdot \Delta T_{steam}} \right]^{1/4} \]

Where the modified latent heat is \( h'_{fg} = h_{fg} + 0.68 \cdot C_{p,liq} \cdot \Delta T_{steam} \).
Wall resistance (\(R_{wall}\)): Based on the plane wall approximation:
\[ R_{wall} = \frac{x_{wall}}{k_{wall}} \]
Food-side film coefficient (\(h_{food}\)): Determined by natural convection correlations using the Rayleigh number (Ra):
\[ Gr = \frac{g \cdot \beta_{food} \cdot \Delta T_{food} \cdot H^3}{\nu^2} \]

\[ Pr = \frac{C_{p,food} \cdot \mu_{food}}{k_{food}} \]

\[ Ra = Gr \cdot Pr \]

\[ h_{food} = \frac{Nu \cdot k_{food}}{H} \]

Where \( Nu = 0.59 \cdot Ra^{1/4} \) for the laminar regime.

Regime/Condition	Criteria
Natural Convection (Laminar)	\( 10^4 < Ra < 10^9 \)
Natural Convection (Turbulent)	\( Ra \geq 10^9 \)
Condensation Film	\( Re_{film} < 1800 \)

To select an accurate fouling factor, process engineers should consider the following criteria:

Consult the TEMA standards for recommended values based on specific fluid types and service conditions.
Evaluate the expected maintenance cycle and the propensity of the process fluid to form deposits or scale.
Account for the velocity of the fluid, as higher velocities can mitigate fouling accumulation.
Review historical performance data from similar existing units within the plant.

The value of U is derived from the thermal resistances of the system. Key variables include:

The individual convective heat transfer coefficients for both the hot and cold fluids.
The thermal conductivity and thickness of the tube wall material.
The resistance introduced by fouling layers on both the internal and external surfaces.
The geometry of the heat exchanger, including fin efficiency if applicable.

Discrepancies between theoretical design and field performance typically arise from:

Actual flow rates deviating from the design basis, leading to incorrect Reynolds numbers.
Unexpected changes in fluid physical properties, such as viscosity or thermal conductivity, due to composition shifts.
Non-uniform flow distribution within the shell or tube passes.
The presence of non-condensable gases in steam-side applications which significantly reduce the effective heat transfer area.

Worked Example: Overall Heat Transfer Coefficient for a Steam Retort

A vertical steel can containing beans in water is processed in a still steam retort. The can height is 0.1 m, wall thickness 0.3 mm, and steel thermal conductivity 50 W/m·K. Steam at 121.0 °C condenses on the outer surface with a wall temperature of 116.0 °C. The food-side fouling resistance is 0.001 m²·K/W. The food has properties similar to water at high temperature.

Knowns

Saturation temperature, \(T_{sat} = 121.0\) °C
Wall temperature, \(T_{wall} = 116.0\) °C
Can height, \(H = 0.1\) m
Condensate liquid density, \(\rho_{liq} = 947.0\) kg/m³
Condensate thermal conductivity, \(k_{liq} = 0.683\) W/m·K
Condensate viscosity, \(\mu_{liq} = 0.000235\) Pa·s
Latent heat of vaporization, \(h_{fg} = 2.202 \times 10^6\) J/kg
Condensate specific heat, \(C_{p,liq} = 4240.0\) J/kg·K
Wall thickness, \(x_{wall} = 0.0003\) m
Steel thermal conductivity, \(k_{steel} = 50.0\) W/m·K
Fouling resistance, \(R_{fouling} = 0.001\) m²·K/W
Food density, \(\rho_{food} = 960.0\) kg/m³
Food thermal conductivity, \(k_{food} = 0.68\) W/m·K
Food viscosity, \(\mu_{food} = 0.00028\) Pa·s
Food thermal expansion coefficient, \(\beta_{food} = 0.00075\) K⁻¹
Food specific heat, \(C_{p,food} = 4200.0\) J/kg·K
Temperature difference for food, \(\Delta T_{food} = 5.0\) °C

Step-by-Step Calculation

Steam-side film coefficient

Compute temperature difference on steam side: \(\Delta T_{steam} = T_{sat} - T_{wall} = 121.0 - 116.0 = 5.0\) °C.

Modified latent heat: \(h_{fg}' = h_{fg} + 0.68 \cdot C_{p,liq} \cdot \Delta T_{steam} = 2.202 \times 10^6 + 0.68 \cdot 4240.0 \cdot 5.0 = 2.216416 \times 10^6\) J/kg.

Using Nusselt correlation for laminar film condensation on a vertical surface:

\[ h_{steam} = 0.943 \cdot \left[ \frac{k_{liq}^3 \cdot \rho_{liq} \cdot (\rho_{liq} - \rho_{vap}) \cdot g \cdot h_{fg}'}{\mu_{liq} \cdot H \cdot \Delta T_{steam}} \right]^{1/4} \]

With \(\rho_{vap} = 0.598\) kg/m³, \(g = 9.81\) m/s², the numerator becomes \(6.208805 \times 10^{12}\) and denominator \(1.175 \times 10^{-4}\), giving a fraction of \(5.284 \times 10^{16}\). Taking the fourth root and multiplying by 0.943 yields:

\[ h_{steam} = 14297.307 \text{ W/m}^2\cdot\text{K} \]

To confirm the laminar film condensation assumption, the film Reynolds number (\(Re_{film}\)) is estimated to be far below the critical value of 1800 for this geometry and fluid properties, validating the use of Nusselt's correlation.
Wall resistance

Conduction resistance: \(R_{wall} = \frac{x_{wall}}{k_{steel}} = \frac{0.0003}{50.0} = 6.0 \times 10^{-6}\) m²·K/W.
Food-side film coefficient

Dynamic viscosity: \(\nu_{food} = \frac{\mu_{food}}{\rho_{food}} = \frac{0.00028}{960.0} = 2.916667 \times 10^{-7}\) m²/s.

Grashof number: \(Gr = \frac{g \cdot \beta_{food} \cdot \Delta T_{food} \cdot H^3}{\nu_{food}^2} = \frac{9.81 \cdot 0.00075 \cdot 5.0 \cdot (0.1)^3}{(2.916667 \times 10^{-7})^2} = 4.324408 \times 10^8\).

Prandtl number: \(Pr = \frac{C_{p,food} \cdot \mu_{food}}{k_{food}} = \frac{4200.0 \cdot 0.00028}{0.68} = 1.729412\).

Rayleigh number: \(Ra = Gr \cdot Pr = 4.324408 \times 10^8 \cdot 1.729412 = 7.478682 \times 10^8\).

For laminar natural convection on a vertical plate (\(Ra < 10^9\)), Nusselt number: \(Nu = 0.59 \cdot Ra^{0.25} = 0.59 \cdot (7.478682 \times 10^8)^{0.25} = 97.568\).

Heat transfer coefficient: \(h_{food} = \frac{Nu \cdot k_{food}}{H} = \frac{97.568 \cdot 0.68}{0.1} = 663.464\) W/m²·K.
Combine resistances

Steam-side resistance: \(R_{steam} = \frac{1}{h_{steam}} = \frac{1}{14297.307} = 6.994 \times 10^{-5}\) m²·K/W.

Food-side resistance: \(R_{food} = \frac{1}{h_{food}} = \frac{1}{663.464} = 1.507 \times 10^{-3}\) m²·K/W.

Total resistance: \(R_{total} = R_{steam} + R_{wall} + R_{fouling} + R_{food} = 6.994 \times 10^{-5} + 6.0 \times 10^{-6} + 0.001 + 1.507 \times 10^{-3} = 2.583 \times 10^{-3}\) m²·K/W.

Overall heat transfer coefficient: \(U_{fouled} = \frac{1}{R_{total}} = \frac{1}{2.583 \times 10^{-3}} = 387.154\) W/m²·K.

Final Answer

The overall heat transfer coefficient for the fouled retort is \(U = 387.2\) W/m²·K (rounded to four significant figures). The dominant resistances are the food-side convection (approximately 58% of total) and the fouling resistance (39%), while the steam-side and wall resistances are negligible.

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