Heat conduction with transient boundary conditions

Example 4.2. Heat Conduction with Transient Boundary Conditions... 

Heat conduction with a constant boundary condition at x =0 was considered in example 4.1. The same technique can be applied for time dependent boundary conditions. Consider the transient heat conduction problem in a slab.[4] The governing equation is ... 

During the molding cooling process, a three-dimensional, cyclic, transient heat conduction problem with convective boundary conditions on the cooling channel and mold base surfaces is involved. The overall heat transfer phenomena is governed by a three-dimensional Poisson equation. 

Problem Solve the one-dimensional, transient heat conduction problem with the following boundary conditions ... 

A spherical metal ball of radius r is heated in an oven to a lempefature of Tj throughout an is then taken out of the oven and dropped into a large body of water at T. where it is coole by convection with an average convection heat transfer coeflicieiit of h. Assuming constant Ihermal conduclivity and transient one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve. 

Consider a semi-inlinite solid with constant thermophysical properties, no internal heat generation, uniform theimal cnnditinn.s on its exposed surface, and initially a uniform temperature of Tj throughout. Heat tfansfec in this case occurs only in the direction uormal to the surface (the x direction), and thus it is one-dimensional. Differential equations are independent of the boundary or initial conditions, and thus Eq. 4—lOa for one-dimensional transient conduction in Cartesian coordinates applies. The depth of the solid is large (x expressed mathematically as a boundary condition as T x —> , 0 = T,. 

This time is reached after 40 steps with M = 1, 20 steps for M = 2, 8 steps for M = 5 and finally 4 steps for M = 10. The temperatures for M = 1 and M = 2 agree very well with each other and with the analytical solution. The values for M = 5 yield somewhat larger deviations, while the result for M = 10 is useless. This large step produces temperature oscillations which are physically impossible. In [2.57], p. 122, a condition for the restriction of the step size, so that oscillations can be avoided, is given for a transient heat conduction problem with boundary conditions different from our example. The transfer of this condition to the present task delivers the limit... 

The basis for the solution of mass diffusion problems, which go beyond the simple case of steady-state and one-dimensional diffusion, sections 1.4.1 and 1.4.2, is the differential equation for the concentration held in a quiescent medium. It is known as the mass diffusion equation. As mass diffusion means the movement of particles, a quiescent medium may only be presumed for special cases which we will discuss first in the following sections. In a similar way to the heat conduction in section 2.1, we will discuss the derivation of the mass diffusion equation in general terms in which the concentration dependence of the material properties and chemical reactions will be considered. This will show that a large number of mass diffusion problems can be described by differential equations and boundary conditions, just like in heat conduction. Therefore, we do not need to solve many new mass diffusion problems, we can merely transfer the results from heat conduction to the analogue mass diffusion problem. This means that mass diffusion problem solutions can be illustrated in a short section. At the end of the section a more detailed discussion of steady-state and transient mass diffusion with chemical reactions is included. 

Figure 4 shows a model of a liquid droplet inside a capillary and under a periodic temperature field. At a relatively low heater temperature, heat radiation can be neglected. The energy equation for heat transport in the capillary wall formulated with heat conduction and free convection remains the same as in the case of a transient temperature field (Eq. 1), but the periodic boundary conditions are now... 

Equation 7.48 with boundary conditions 7.49 is simply the equation for transient conductive heat transfer in a cylinder, where z is the timelike variable, except that Bi varies with z. We know that Bi is relatively insensitive to the velocity and can be approximated by a constant value in the lower portion of the spudine, where little attenuation occurs. The solution to Equations 7.48 and 7.49 for constant Bi is... 

Here, the concentration at x = 0 always remains constant contrary to the previous example, where a fixed concentration is introduced once. This problem is analogous to transient conduction in semi-infinite solid with constant surface temperature boundary condition. The detailed solution procedure can be found in regular heat transfer book. The solution for the above problem can be obtained by using f The governing equation in partial differential form... 

In case of steady state heat conduction, the material property is the conductivity which can be calculated once the heat loss from the body is known and the boundary temperature is measured. In case of transient heat flow, the main factor is the diffusivity a which is equal to the ratio of the conductivity and the heat content of the body. Hence equation 10.8 is the basic governing equation of transient heat transfer with boundary conditions relevant for textile materials. Transient state heat conduction is related to instantaneous conduction of heat fi-om the surface of the body to the clothing. Instantaneous heat transfer can be related to the warmth or coolness to touch and the warm-cool feeling of any clothing can be quantified. 

Sensible heat and the latent heat of freeing are removed from the water at the liquid/solid interface. Under the prevailing static conditions heat will pass from the water to the cold sink by conduction. The resistance to this heat flow initially, will be a combination of the thermal resistances in the liquid and due to the cold solid. Immediately a layer of ice be ns to form on the cold surface a further resistance is added to the other thermal resistances. As further heat is extracted from the water, the ice layer thickens representing an advancing boundary between the solid ice and the liquid water, i.e. a transient condition. The transient condition, coupled with complex geometries and different forms of ice structure dependent in turn on the rate of cooling, constitute severe problems of mathematical analysis. 

The gas is injected from the bottom with a temperature of 263.15 K (—10 °C) and the initial particle temperature was 233.15 K (—40 °C). Consequently, the particle will be eventually heated up to the gas temperature. The simulations were conducted by a three-step procedure. Firstly, a steady-state flow field about the particle was calculated with constant air temperature. Then the temperature field was calculated without flow, so that only a thermal boundary layer develops around the cold particle. These results were used as the initial condition for the transient coupled simulation of flow field and temperature. 

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