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Heat transfer
| Content Provider | Scilit |
|---|---|
| Author | Alawadhi, Esam M. |
| Copyright Year | 2015 |
| Description | Book Name: Finite Element Simulations Using ANSYS |
| Abstract | Whenever a temperature gradient exists in a solid, heat will ow from the high-temperature region to the low-temperature region. The basic governing heat conduction equation is obtained by considering a plate with a surface area A and thickness Δx, as shown in Figure 5.1. One side is maintained at temperature T1 and the other side is at temperature T2. Experimental observation indicates that the rate of heat ow is directly proportional to the area and temperature difference, but inversely proportional to the plate thickness. The proportionality sign is replaced by an equal sign by introducing the constant k as follows: Q kA T T = − ∆ (5.1) The constant k is the thermal conductivity of the plate. This property depends on the type of the plate’s material. Equation 5.1 is also called Fourier’s law. Fourier’s law can be expressed in differential form in the direction of the normal coordinate: Q kA dT dn = − (5.2) Also, Fourier’s law can be expressed for multidimensional heat ux ow as follows: q k T x iˆ T y jˆ T z kˆ= − ∂ ∂ + ∂ ∂ + ∂ ∂ (5.3) An energy balance can be applied to a differential volume dx dy dz, for conduction analysis in a Cartesian coordinate, as shown in Figure 5.2. The objective of this energy balance is to obtain the temperature distribution within the solid. The temperature distribution is used to determine the heat ow at a certain surface, or to study the thermal stress. The heat ux perpendicular to the surface of the control volume is indicated by the terms, qx, qy, and qz. The heat ux at the opposite surface can then be expressed using the Taylor series expansion of the rst order as follows: q q q x = + ∂ ∂+ (5.4) q q q y = + ∂ ∂+ (5.5) q q q z = + ∂ ∂+ (5.6) Energy can be generated in the medium, and the expression of the heat generation is E q dx dy dzgen = (5.7) where q is the generated heat per unit volume, W/m3. If the heating process is unsteady, the total energy of the control volume can be increased or decreased. The energy storage term is expressed as E C T t dx dy dzst P = ρ ∂ ∂ (5.8) The sum of the energy generation in the control volume and net heat ow should be equal to the energy stored in the control volume. The energy conservation can be expressed in the following mathematical form: E E E Egen in out st( )+ − = (5.9) Substituting expressions (5.4-5.8) into (5.9), the energy conservation equation becomes + ∂ ∂ + ∂ ∂ + ∂ ∂ = ρ ∂ ∂q dx dy dz q x dy dz q y dx dz q z dx dy C T t (5.10) The Qx′′, Qy′′, and Qz′′ are obtained from Fourier’s law (5.2) as follows: Q k dT dx dxx′′ = − (5.11) Q k dT dy dyy′′ = − (5.12) Q k dT dx dzz′′ = − (5.13) Finally, the conduction energy equation per unit volume, in a Cartesian coordinate, is expressed as ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + = ρ ∂ ∂x k T x y k T y z k T z q C T t p (5.14) When the system reaches the steady-state condition, the term ∂T/∂t is equal to zero. If the thermal conductivity is independent of the direction, the conduction energy equation can be written in a simpler form as ∂ ∂ + ∂ ∂ + ∂ ∂ + = ρ ∂ ∂ T x T y T z q k C k T t The energy equation is a partial differential equation with second order in space and rst order in time. The boundary conditions along its surface as well as the initial condition must be specied. For the initial condition, the temperature distribution in the system must be provided. In heat transfer problems, there are three types of boundary conditions: temperature, heat ux, and convection. |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2014-0-31662-5&isbn=9780429067839&doi=10.1201/b18949-9&format=pdf |
| Ending Page | 306 |
| Page Count | 68 |
| Starting Page | 239 |
| DOI | 10.1201/b18949-9 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2015-09-18 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Finite Element Simulations Using Ansys Thermodynamics Thermal Conductivity Heat Differential Temperature Distribution Cartesian Coordinate Rst Order Dy Dz |
| Content Type | Text |
| Resource Type | Chapter |
