Transport Phenomenon

Transport is the movement of a physical quantity driven by a force

Naturally, we would be interested in answering questions like

• What is being transported?
• How much?
• Rate of transfer?
• Can we alter the rate of transfer?

In the below illustration, we can observe that heat energy supplied by the stove is absorbed by the dissolved gases in water resulting in the rising of bubbles as shown. These gas molecules serve as carrier and the energy is transported from the base of the stove at a higher temperature to the ambient at a lower temperature.

Major Applications

Major Applications

Understanding Natural Phenomenon

• Climate Change
• Generation of dew and fog
• Heating and cooling of earth surfaces
• Formation of rain and snow

Major Applications

Industrial Applications

• Heat Exchangers
• Manufacturing Technology
• Automobiles
• Aircraft
• Power Plants
• Electronics cooling

Modes of Heat Transfer

Energy transfer owing to temperature difference is called heat transfer. The transport of thermal energy can happen in three different modes

• Conduction
• Convection
• Radiation

In reality, all three can be simultaneously present with different relative strengths

where \(q''\) is heat flux, \(k\) is thermal conductivity, \(h\) is the heat transfer coefficient, \(epsilon\) is the emissivity of radiative surface, \(\sigma\) is Stefan-Boltzmann constant (=\(5.67 \times 10^{-8} W/m^2K^4\)), \(T_w,T_0\) are the surface and ambient temperatures respectively.

Convection vs Conduction

Convection = Conduction + Advection

Conduction is a diffussion process. By diffusion, we mean the particles/carriers of thermal energy do not have any preferential direction of movement. However, the extent of this movement is proportional to the energy of the carriers. The carriers at higher temperature have higher energy. The differential rates of these movement appears as the transport of energy in a direction from higher to lower.

Advection is an active transport process. The carriers transport the thermal energy through their momentum which in turn could be generated by an external pressure field. Hence, such a transport will have a preferential direction for movement of carriers.

In general, both these processes exist simultaneously, and we consider a superposition of these two effects in the upcoming classes.

Classification of flows

Classification of flows

External vs Internal Flows

Classification of flows

Forced, Mixed and Natural Convection Flows

In forced convection, we have a fan or blower or pump to drive the flow. This driver is present near the boundaries of our region of interest.

In natural convection, we have body forces like buoyancy driving the flow. A buoyancy source will be present in our region of interest.

Even though both driving forces are present, the effect on our object of interest (usually the net heat flux from a surface) might not be strongly related with all the driving forces. In upcoming sessions, we will look at a way to estimate the relative strengths of these forces on the net heat transfer.

If both the forces exert similar orders of effect, then we consider that a mixed convective flow.

Consider the smoke coming out of a chimney as shown below. The slider position is proportional to the wind speed. Thus, increasing the slider increases the strength of forced convection. If you were to draw a free body diagram for the particle as it travels along, how would the plots of horizontal and vertical forces on the particle look like?

Incompressible Flow Equations

Incompressible Flow Equations

The governing equations for an unsteady, incompressible, newtonian and constant properties fluid are given in their vector form as

Continuity: \(\vec{\nabla}.\vec{v} = 0\)

Momentum: \(\rho \left(\frac{\partial \vec{v}}{\partial t} + \vec{v}.\vec{\nabla} \vec{v} \right) = -\vec{\nabla p} + \mu \nabla^2 \vec{v} + \rho \vec{g}\)

where
\(\vec{v} = u \hat{i} + v \hat{j} + w \hat{k}\) is the velocity field
\(p\) is the pressure field
\(\rho\) is the density of the fluid
\(\mu\) is the kinematic viscosity of the fluid
\(\vec{g}\) is the acceleration due to gravity
\(t\) is the time

Incompressible Flow Equations

For a 3D cartesian co-ordinate system, the equations are as follows

Continuity: \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}=0\)

X-momentum: \(\rho \left( \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} \right) = -\frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} +\frac{\partial^2 u}{\partial z^2} \right) + \rho g_x\)

Y-momentum: \(\rho \left( \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} \right) = -\frac{\partial p}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} +\frac{\partial^2 v}{\partial z^2} \right) + \rho g_y\)

Z-momentum: \(\rho \left( \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} \right) = -\frac{\partial p}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} +\frac{\partial^2 w}{\partial z^2} \right) + \rho g_z\)

Observe that the continuity equation does not have any time derivative. This is because of our assumption where the properties are constant. However, it is interesting to note that for low speed flows the above equation still holds Babu, V. (2022).

Incompressible Flow Equations

Y-momentum: $$ \rho \underbrace{\frac{\partial v}{\partial t}}_{\text{unsteady}} + \rho \underbrace{\left( u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} \right)}_{\text{advection}} = \underbrace{-\frac{\partial p}{\partial y}}_{\text{pressure}} + \underbrace{\mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right)}_{\text{viscous}} + \underbrace{\rho g_y}_{\text{gravity}} $$

Incompressible Flow Equations

Boundary Conditions

• Dirichlet: a constant value of the quantity is specified
• Neumann: the gradient is specified
• Mixed: a linear combination of the above two is specified