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Conduction: One Dimensional, Heat Transfer
General heat conduction relation in cylindrical coordinate system (fig. 2.12) is derived (briefly) below.
Fig.2.12. Cylindrical coordinate system (a) and an element of the cylinder
The energy conservation for the system is written as,
Ӏ + ӀӀ = ӀӀӀ + ӀV (2.21)
where,
I : Rate of heat energy conducted in
II : Rate of heat energy generated within the volume element
III : Rate of heat energy conducted out
IV : Rate of energy accumulated (ӀV)
and the above terms are defines as,
Thus,
On putting the values in equation 2.21,
Thus the Laplacian operator is,
(a) (b)
Fig.2.13. Spherical coordinate system (a) and an element of the sphere
In a similar way the general expression for the conduction heat transfer in spherical body with heat source can also be found out as per the previous discussion. The Laplacian operator for the spherical coordinate system (fig.2.13) is given below and the students are encouraged to derive the expression themselves.
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FAQs on Conduction: One Dimensional - 6 - Heat Transfer - Mechanical Engineering
| 1. What is conduction in one dimension? |  |
| 2. How does conduction in one dimension differ from conduction in multiple dimensions? | |
Ans. Conduction in one dimension differs from conduction in multiple dimensions in terms of the direction of heat transfer. In one dimension, heat transfer occurs in a straight line or in a single direction, whereas in multiple dimensions, heat can flow in multiple directions simultaneously.
| 3. What factors affect the rate of conduction in one dimension? | |
Ans. The rate of conduction in one dimension is influenced by several factors, including the temperature difference between the two ends of the object, the thermal conductivity of the material, the cross-sectional area of the object, and the length of the object.
| 4. How is the rate of conduction calculated in one dimension? | |
Ans. The rate of conduction in one dimension can be calculated using Fourier's law of heat conduction. It states that the rate of heat transfer (Q) is directly proportional to the product of the cross-sectional area (A), the temperature difference (ΔT), and the thermal conductivity (k), and inversely proportional to the length (L) of the object. Mathematically, it can be represented as Q = kA(ΔT/L).
| 5. What are some practical applications of conduction in one dimension? | |
Ans. Conduction in one dimension has various practical applications. It is utilized in the design and operation of heat exchangers, refrigeration systems, and thermal insulation materials. It is also important in the field of chemical engineering for optimizing heat transfer processes in reactors, distillation columns, and other industrial equipment.
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