3. Convection

3.1 Introduction

Three factors play major roles in convection heat transfer: (i) fluid motion, (ii) fluid nature, and (iii) surface geometry.

Using experimental observations, it is postulated that surface flux in convection is directly proportional to the difference in temperature between the surface and the streaming fluid. That is, \[q \propto (T_s-T_\infty)\] \(T_s\) is surface temperature and \(T_\infty\) is the fluid temperature far away from the surface. Introducing a proportionality constant to express this relationship as equality, we obtain \[q = h(T_s-T_\infty)\] This result is known as Newton’s law of cooling. The constant of proportionality \(h\) is called the heat transfer coefficient.

Unlike thermal conductivity \(k\), the heat transfer coefficient is not a material property. Rather it depends on geometry, fluid properties, motion, and in some cases temperature difference, \(\Delta T=(T_s-T_\infty)\). That is \[h = f(\text{geometry, fluid motion, fluid properties}, \Delta T)\] \(h\) is related with \(k\) as, \[h = \frac{-k\left.\dfrac{\partial T}{\partial y}\right|_{y=0}}{T_s-T_\infty}\]

\(\displaystyle \left.\dfrac{\partial T}{\partial y}\right|_{y=0}\) depends on the whole fluid motion, and both fluid flow and heat transfer equations are needed to find this.

Flow Generation

Since fluid motion is central to convection heat transfer, we will be concerned with two common flow classifications:

• Forced convection: Fluid motion is generated mechanically through the use of a fan, blower, nozzle, jet, etc.. Fluid motion relative to a surface can also be obtained by moving an object, such as a missile, through a fluid.

• Free (natural) convection: Fluid motion is generated by gravitational field. However, the presence of a gravitational field is not sufficient to set a fluid in motion. Fluid density change is also required for free convection to occur. In free convection, density variation is primarily due to temperature changes.

  Fluid motion is confined inside the thermal boundary layer. The velocity is zero both at the wall and far away from it.

Significant Parameters in Convective Heat Transfer

Since a lot of variables influence convective heat transfer, the variables conveniently grouped in to dimensionless numbers, by dimensional analysis.

The molecular diffusivities of momentum and energy (heat) have been defined previously as \[\begin{aligned} \text{momentum diffusivity:} \qquad \nu &= \frac{\mu}{\rho} \\ \text{thermal diffusivity:} \qquad \alpha &= \frac{k}{\rho C_P} \end{aligned}\] Both have same dimensions \(L^2/t\); thus their ratio must be dimensionless. This ratio, that of molecular diffusivity of momentum to the molecular diffusivity of heat, is designed the Prandtl number. \[\text{Pr} = \frac{\nu}{\alpha} = \frac{C_P\mu}{k}\] Prandtl number is observed to be a combination of fluid properties; thus Pr itself may be thought of as a property. Primarily it is a function of temperature.

Prandtl Number of Typical Fluids

Variables of Importance in Forced Convection

Variables: \(h, D, v,\rho,\mu, k, C_P\).

By dimensional analysis, \[\text{Nu} = \phi(\text{Re}, \text{Pr})\]

Variables of Importance in Natural Convection

Requirements:

• Gravitational field

• Density change with temperature

Variables: \(h, L, \rho,\mu, k, C_P, \beta, g, \Delta T\)

\(\beta\) is the coefficient of thermal expansion. \[\beta = \frac{1}{V}\left(\frac{\partial V}{\partial T} \right)_P = -\frac{1}{\rho}\left(\frac{\partial \rho}{\partial T} \right)_P\] For ideal gases, \(\beta=1/T\)

: \[\text{Nu} = \phi(\text{Gr}, \text{Pr})\]

: \[\text{Gr} = \frac{g \beta \rho^2L^3(T_w-T_{\infty})}{\mu^2}\]

The role played by Reynolds number in forced convection is played by the Grashof number in natural convection. The critical Grashof number is observed to be about \(10^9\) for vertical plates.