1) Introductions for the heat and mass transfer

Why the mechanical engineer should know how the heat and mass is transferred? for mechanical engineers, it is essential for them to analyze:

• how the reaction between the mechanical components
• how the individual component or subsystem endures long for the given conditions
• how much mechanical power transmission occurs by heat and mass transfer

However, we don’t know exactly what concept we need to take the heat and mass transfer. So, let us talk together what preliminaries are in need.

More detailed concepts would be found in thermodynamics post, so we just remind the things briefly.

1. Nomenclatures

• System, surroundings and boundary:

  • System: The space of $n$-th dimension, which contains all things what we want to analyze.
  • Surroundings: Everything excluding the system, whose dimension is also $n$-th one.
  • Boundary: The $(n-1)$-th-dimensioned distinguisher between system and surroundings.
  • Mainly we’re involved in $n\,=\,3$

  e.g.) Control volume $\rightarrow$ system Whole world excluding system $\rightarrow$ surroundings Control surface $\rightarrow$ boundary

• Closed and Isolated System

  • Closed system: the system which always contain the same matter - no heat and mass transfer across its boundary.
  • Isolated system: a special type of closed system that does not interact in any way with its surroundings

• Heat transfer (or Heat)

  • Thermal energy in transit due to a spatial temperature difference.
  • Surface transfer phenomena There are three types of heat transfer: conduction, convection and radiation

Fig. 1. How the conduction, convection and radiation occurs - brief diagram

• Transport Phenomena
  • There are several topics for the transport phenomena:
  • Three levels at which transport phenomena can be studied
    • Macroscopic level - the equations for the macroscopic balance(mass, momentum, energy, and angular momentum), no attempt for the detailed stuffs
    • Microscopic level - the equations of change of above macroscopic values in microscopic region.
    • Molecular level - we seek a fundamental understanding of the mechanisms of mass, momentum, energy and angular momentum transport in terms of molecular structure and intermolecular forces.

Fig. 2. Topics of transport phenomena

Brief concepts of each the conduction, convection and radiation are below:

2. Conduction

• Atomic and molecular activity: processes at these levels sustain this mode of heat transfer
• Caused by the direct contact between the objects containing heat energy
• Physical mechanism of conduction: most easily explained by considering a gas and using ideas (Familiar from thermodynamics background)
• In the presence of a temperature gradient, energy transfer by conduction must then occur in the direction of decreasing temperature.
• The net transfer of energy by random molecular motion as a diffusion of energy.
• Heat transfer processes can be quantified in terms of appropriate rate equations.

Fig. 3. Association of conduction heat transfer with diffusion of energy (molecular)

Fig. 4. 1-dimensional heat transfer

• The heat flux $q_x’’\,=\,-k \frac{dT}{dx}$ [Unit: $\frac{W}{m^2}$] where $q_x’’$ is the heat transfer rate in the $x$-direction per unit area perpendicular to the direction of transfer.
• The heat rate $q_x\,=\,q_x’’\,\cdot A$ [W]
• The steady-state conditions (where the temperature distribution is linear), the temperature gradient may be expressed as $\frac{dT}{dx}\,=\,\frac{T_2\,-\,T_1}{L}$. (note the [Fig. 4.1] the meaning of each symbol) so, heat flux is expressed like $q_x’’\,=\,k\frac{T_1\,-\,T_2}{L}\,=\,k\frac{\Delta T}{L}$

3. Convection

Due to random molecular motion and (diffusion), energy is also transferred by the bulk, or macroscopic, motion of the fluid.

• Convection: customarily used when referring to this cumulative transport
• Advection: transport due to bulk fluid motion.
• Hydrodynamic Boundary Layer: The development of a region in the fluid through which the velocity varies from zero at the surface to a finite value $u_{\infty}$ associated with the flow, (consequence of the fluid–surface interaction.)
• Thermal Boundary Layer: A region of the fluid through which the temperature varies from at to in the outer flow as the surface and flow temperatures differ.
  • may be smaller, larger, or the same size as that through which the velocity varies.

Fig. 5. Boundary layer development in convection heat transfer.

• Forced convection: The flow is caused by external means. e.g.) a fan, a pump, or atmospheric winds.

• Natural convection: For free (or natural) convection, the flow is induced by buoyancy forces, which are due to density differences caused by temperature variations in the fluid.

• **The most frequent form of heat transfer.**

• In addition to energy transfer due to random molecular motion (diffusion), energy is also transferred by the bulk, or macroscopic, motion of the fluid.

• The appropriate rate equation: $q’’\,=\,h\begin{vmatrix}T_s\,-\,T_{\infty}\end{vmatrix}$. where $q’’$ is the convective heat [$W/m^2$], $T_s$: surface temperature, $T_{\infty}$: fluid temperature

4. Radiation

Energy emitted by matter that is at a nonzero temperature. Regardless of the form of matter, the emission may be attributed to changes in the electron configurations of the constituent atoms or molecules. The radiation rate equation is below:

$$q''_{rad}\,=\,\frac{q}{A}\,=\,\underset{radiation}{\underline{\epsilon E_b(T_s)}}\,-\underset{absorption}{\underline{\alpha G\,}}=\,\underset{the\,net\,rate\,of\,heat\,flux}{\underline{\epsilon\,\sigma(T_s^4\,-\,T_{sur}^4)}}$$

where $q\prime\prime_{rad}$: the rate of radiation, $E_b$: the emissive power of a body $\epsilon$: emissivity of a gray body, such $0≤\epsilon≤$1, $\sigma$: Stefan Boltzman Constant $(\sigma\,=\,5.67\,\times\,10^{-8}\,W/m^2\,\cdot\,K)$, $G_{abs}\,=\alpha\,G$ where $\alpha\,=\,$absoptity, $0≤\alpha≤1$. Let me explain each single term.

Firstly, the emissive power in ‘ideal’ case is $E_b\,=\,\sigma\,T_s^4$. This is the consideration of black body, and the real emissive power is that of gray body, which is smaller emissive power than that of black body. To make up this differences, the quantity named of “emissivity” is applied, so this is the change: $E\,=\,\epsilon \sigma T_s^4$.

Secondly, let’s talk about the [Fig. 6.] Surface doesn’t absorb a whole radiation. it reflects the radiation, so this phenomena is called as “irradiation”. A portion of the irradiation may be absorbed by the surface. To find out the net rate of the radiation, it must also be considered. the absorbed quantity of irraditation is like: $G_{abs}\,=\,\alpha\,G$

Combining these two term, the above net rate equation is constructed.

Fig. 6. Radiation Exchange

From now on, we’ll study three forms of heat transfer more deeply. Hope that this material would be help for your journey to the heat and mass transfer.

• References
  • Frank P. Incropera, “Fundamentals of Heat and Mass Transfer”, 7th ed.
  • R. Byron Bird, Warren E. Stewart, “Transport Phenomena”, 2nd ed.
  • Yunus A. Cengel, “Thermodynamics - An Engineering Approach”, 8th ed.

Twitter  facebook  Google+ 

Contact

Name: * Email Address: * Message: *
* indicates a required field