5.2 Natural & Forced Convection Heat Transfer Correlations

5.2 Natural & Forced Convection Heat Transfer Correlations

Convective heat transfer occurs between a solid surface and an adjacent moving fluid (liquid or gas) at different temperatures. Convection is classified into forced convection, where fluid motion is driven by external mechanical means (pumps, fans, blowers, or wind), and natural (free) convection, where fluid motion is induced by buoyancy forces arising from density gradients caused by temperature variations in a body force (gravitational) field.

---

1. Newton's Law of Cooling & Boundary Layer Fundamentals

The fundamental rate equation for convective heat transfer is Newton's Law of Cooling:

$$q = h A_s (T_s - T_\infty)$$

Where:

• $q$ = Convective heat transfer rate ($\text{W}$)
• $h$ = Convective heat transfer coefficient ($\text{W/m}^2\cdot\text{K}$ or $\text{W/m}^2\cdot^\circ\text{C}$)
• $A_s$ = Surface area in contact with fluid ($\text{m}^2$)
• $T_s$ = Solid surface temperature ($\text{K}$ or $^\circ\text{C}$)
• $T_\infty$ = Free-stream (or bulk fluid) temperature ($\text{K}$ or $^\circ\text{C}$)

Boundary Layer Theory

When a fluid flows over a solid surface, fluid particles adjacent to the wall adhere due to viscous forces, creating a no-slip condition ($u(y=0) = 0$).

1. Velocity (Hydrodynamic) Boundary Layer ($\delta$): Region of fluid flow where velocity gradients exist normal to the surface, defined as the distance from wall $y = \delta$ where $u(y) = 0.99 U_\infty$.
2. Thermal Boundary Layer ($\delta_t$): Region where fluid temperature gradients exist, defined as distance $y = \delta_t$ where $\frac{T(y) - T_s}{T_\infty - T_s} = 0.99$.

The relative thickness of velocity and thermal boundary layers is governed by the Prandtl number ($Pr$):

$$\frac{\delta}{\delta_t} \approx Pr^n \quad (n \approx 1/3$$

---

2. Key Dimensionless Groups in Convective Heat Transfer

To standardize empirical correlations independent of fluid type or physical scale, heat transfer analysis relies on dimensionless numbers:

Volumetric Thermal Expansion Coefficient ($\beta$)

For ideal gases, the thermal expansion coefficient $\beta$ is evaluated using absolute film temperature $T_f$ in Kelvin:

$$\beta = \frac{1}{T_f} = \frac{1}{\frac{T_s + T_\infty}{2} + 273.15} \quad [\text{K}^{-1}]$$

---

3. Forced Convection Correlations for Internal & External Flows

Forced Flow Inside Tubes and Pipes (Internal Flow)

Flow regime transition in smooth circular pipes occurs at critical Reynolds number $Re_{D,c} \approx 2300$.

• Laminar Fully Developed Flow ($Re_D < 2300$):

  • Constant Surface Temperature ($T_s = \text{const}$): $Nu_D = \frac{h D}{k_f} = 3.66$
  • Constant Surface Heat Flux ($q'' = \text{const}$): $Nu_D = \frac{h D}{k_f} = 4.36$

• Turbulent Fully Developed Flow ($Re_D > 10,000, 0.6 \le Pr \le 160, L/D \ge 10$): Dittus-Boelter Correlation:

  $$Nu_D = 0.023 Re_D^{0.8} Pr^n$$

  Where:

  • $n = 0.4$ for heating of the fluid ($T_s > T_b$)
  • $n = 0.3$ for cooling of the fluid ($T_s < T_b$)
  • All fluid properties are evaluated at the bulk mean temperature $T_b = \frac{T_{in} + T_{out}}{2}$.

• Sieder-Tate Correlation (for large property variations due to steep temperature gradients):

  $$Nu_D = 0.027 Re_D^{0.8} Pr^{1/3} \left( \frac{\mu_b}{\mu_s} \right)^{0.14}$$

External Flow Over Flat Plates

Critical transition Reynolds number for flat plate flow is $Re_{x,c} \approx 5 \times 10^5$.

• Laminar Flow ($Re_x < 5 \times 10^5$):

  • Local Nusselt Number: $Nu_x = 0.332 Re_x^{1/2} Pr^{1/3}$
  • Average Nusselt Number over length $L$: $\overline{Nu}_L = 0.664 Re_L^{1/2} Pr^{1/3}$

• Mixed / Combined Boundary Layer ($Re_L > 5 \times 10^5$):

  $$\overline{Nu}_L = (0.037 Re_L^{4/5} - 871) Pr^{1/3}$$

External Flow Across Single Cylinders

For cross-flow across a circular cylinder of diameter $D$, the Churchill-Bernstein Correlation applies across all $Re_D Pr > 0.2$:

$$\overline{Nu}_D = 0.3 + \frac{0.62 Re_D^{1/2} Pr^{1/3}}{\left[ 1 + (0.4/Pr)^{2/3} \right]^{1/4}} \left[ 1 + \left( \frac{Re_D}{282,000} \right)^{5/8} \right]^{4/5}$$

---

4. Natural (Free) Convection Correlations & Flow Regimes

In free convection, fluid motion is governed by buoyancy-driven Rayleigh number ($Ra_L = Gr_L \cdot Pr$). Standard correlations take the general functional form:

$$\overline{Nu}_L = C Ra_L^n$$

• Laminar Regime ($10^4 \le Ra_L \le 10^9$): Typically $C \approx 0.59$ and $n = 1/4 = 0.25$.
• Turbulent Regime ($10^9 < Ra_L \le 10^{13}$): Typically $C \approx 0.10$ and $n = 1/3 \approx 0.333$.

Vertical Plates and Cylinders

For isothermal vertical surfaces over the entire range of Rayleigh numbers, Churchill and Chu Correlation is widely utilized:

$$\overline{Nu}_L = \left\{ 0.825 + \frac{0.387 Ra_L^{1/6}}{\left[ 1 + (0.492/Pr)^{9/16} \right]^{8/27}} \right\}^2$$

Horizontal Plates

For a horizontal plate of area $A$ and perimeter $P$, characteristic length is defined as $L_c = \frac{A}{P}$.

• Upper Surface Heated (or Lower Surface Cooled):

  • Laminar ($10^4 \le Ra_L \le 10^7$): $\overline{Nu}_L = 0.54 Ra_L^{1/4}$
  • Turbulent ($10^7 < Ra_L \le 10^{11}$): $\overline{Nu}_L = 0.15 Ra_L^{1/3}$

• Lower Surface Heated (or Upper Surface Cooled):

  • Laminar ($10^5 \le Ra_L \le 10^{10}$): $\overline{Nu}_L = 0.27 Ra_L^{1/4}$

---

5. Mixed Convection & Regime Identification ($Gr_L / Re_L^2$)

When both external flow forcing and internal thermal buoyancy forces coexist, the relative dominance of heat transfer mode is evaluated by comparing Grashof and Reynolds numbers:

$$\frac{Gr_L}{Re_L^2} = \frac{g \beta (T_s - T_\infty) L}{\frac{V^2}{1}}$$

1. $\frac{Gr_L}{Re_L^2} \ll 1$: Pure forced convection; buoyancy effects are negligible.
2. $\frac{Gr_L}{Re_L^2} \gg 1$: Pure natural convection; forced flow velocity is negligible.
3. $\frac{Gr_L}{Re_L^2} \approx 1$: Mixed Convection; both forced and natural convection contributions must be calculated using combined Nusselt relations: $Nu_{mixed}^n = Nu_{forced}^n \pm Nu_{natural}^n$ ($n \approx 3$).

---

6. Step-by-Step Worked Engineering Calculation (MELE Board Exam Style)

Problem Statement

Water at a bulk mean temperature of $T_b = 40^\circ\text{C}$ flows inside a smooth copper pipe with an internal diameter of $D = 25\text{ mm} = 0.025\text{ m}$ and length $L = 4.0\text{ m}$ at a mean velocity of $V = 2.0\text{ m/s}$. The inner pipe wall surface is maintained at a uniform temperature of $T_s = 80^\circ\text{C}$ by external condensing steam.

Water properties at $T_b = 40^\circ\text{C}$:

• Density $\rho = 992.1\text{ kg/m}^3$
• Dynamic viscosity $\mu = 0.653 \times 10^{-3}\text{ N}\cdot\text{s/m}^2$
• Thermal conductivity $k_f = 0.631\text{ W/m}\cdot\text{K}$
• Prandtl number $Pr = 4.32$

1. Calculate the Reynolds number $Re_D$ and verify the flow regime.
2. Calculate the convective heat transfer coefficient $h$ using the Dittus-Boelter correlation.
3. Determine the total heat transfer rate $q$ to the water.

Solution

Step 1: Calculate Reynolds Number & Identify Flow Regime

$$Re_D = \frac{\rho V D}{\mu} = \frac{992.1 \times 2.0 \times 0.025}{0.653 \times 10^{-3}} = \frac{49.605}{0.000653} = 75,964.8$$

Analysis: Since $Re_D = 75,965 > 10,000$, the internal flow is fully turbulent.

Step 2: Calculate Nusselt Number & Convective Coefficient

Since water is being heated ($T_s = 80^\circ\text{C} > T_b = 40^\circ\text{C}$), the Prandtl number exponent in Dittus-Boelter equation is $n = 0.4$:

$$Nu_D = 0.023 Re_D^{0.8} Pr^{0.4}$$

Calculate terms:

• $Re_D^{0.8} = (75,964.8)^{0.8} = 8057.06$
• $Pr^{0.4} = (4.32)^{0.4} = 1.7919$

Substitute into Dittus-Boelter:

$$Nu_D = 0.023 \times 8057.06 \times 1.7919 = 332.09$$

Now calculate convective heat transfer coefficient $h$:

$$h = \frac{Nu_D \cdot k_f}{D} = \frac{332.09 \times 0.631\text{ W/m}\cdot\text{K}}{0.025\text{ m}} = \frac{209.549}{0.025} = 8381.96\text{ W/m}^2\cdot\text{K}$$

Step 3: Calculate Internal Surface Area and Total Heat Rate

Internal surface area of pipe $A_s$:

$$A_s = \pi D L = \pi \times 0.025 \times 4.0 = 0.31416\text{ m}^2$$

Calculate convective heat transfer rate $q$ using Newton's Law of Cooling:

$$q = h A_s (T_s - T_b) = 8381.96 \times 0.31416 \times (80 - 40)$$ $$q = 8381.96 \times 0.31416 \times 40 = 105,327.5\text{ W} = 105.33\text{ kW}$$

The total convective heat transfer rate from pipe wall to water is $105.33\text{ kW}$.