Example Transient Heat Transfer

The next example is a transient heat transfer problem. 

Steps 1-3 You make the same FEMLAB choices as for heat conduction. 

Step 4 In the Physics/Subdomain Settings mode the equation listed at the top is 

Step 5 In this same mode, you choose the Init tab, and set the initial conditions to 1.0. 

Step 6 In the Physics/Boundary Settings mode, you set boundary 1 (the left-hand side) to Insulated and boundary 2 (the right-hand side) to Temperature, and set the value to 0. 

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In this section, we instead consider two well-known examples of heat transfer in the fully developed, laminar, and unidirectional flow of a Newtonian fluid in a straight circular tube. We begin with a problem in which there is a prescribed heat flux into the fluid at the walls of the tube, so that there is a steady-state temperature distribution in the tube. At the end of the section, we consider the transient evolution of the temperature distribution beginning with an initially sharp temperature jump within the fluid at a fixed position (say z = 0), which illustrates an important phenomenon that is known as Taylor dispersion. 

Problem 4.9. Heat transfer obeys the same basic laws as diffusion. For example, a spherical transient heat transfer process can be described with the following equation ... 

For a transient heat transfer process, for example, for heating up or cooling down a body, we have to consider the variation of temperature with time as well as with position. For a large plane wall of thickness 21, the heat conduction perpendicular to the (almost infinite) area A of the plate is one-dimensional. To derive the respective differential equation, we use the energy balance for a small slice with thickness Ax and volume A Ax (Figure 3.2.19) ... 

For calculations of the transient heat transfer by conduction and convection, for example, heating up a body, the Fourier s second law is used, and we need to know the Fourier number Fo and Biot number Bi. The advantage of these numbers is that we can use charts, which depict the dimensionless temperature for given values of Fo and Bi. 

For overpower transients that are fast with respect to the fuel rod thermal time constant (for example, the uncontrolled RCCA bank withdrawal from subcritical or lower power startup and RCCA ejection incident, both of which result in a large power rise over a few seconds), a detailed fuel transient heat transfer calculation is performed. 

Dimensionless analysis is a powerfiil tool in analyzing the transient heat transfer and flow processes accompanying melt flow in an injection mold or cooling in blown film,to quote a couple of examples. However, because of the nature of non-Newtonian polymer melt flow the dimensionless mmibers used to describe flow and heat transfer processes of Newtonian flnids have to be modified for polymer melts. This paper describes how an easily applicable equation for the cooling of melt in a spiral flow in injection molds has been derived on the basis of modified dimensionless numbers and verified by experiments. Analyzing the air gap dynamics in extrusion coating is another application of dimensional analysis. 

Steady state heat transfer refers to the condition where the rate of heat flowing into one face of an object is equal to that flowing out of the other. If, for example, a slab of metal were placed on a hot-plate, the heat flowing into the metal would initially contribute to a temperature rise in the material, until ultimately a linear temperature gradient formed between the hot and cold faces, wherein heat flowing in would equal heat flowing out and steady state heat transfer would be established. The time involved before steady state conditions axe encountered is dependent on the thermal requirements, that is, the total heat capacity of the material. A useful constant, therefore, in depicting transient, or non-steady state heat transfer is the thermal diffusivity ... 

Most heat transfer problems encountered in practice are transient in nature, but they are usually analyzed under some presumed steady conditions since steady processes are easier to analyze, and they provide the answers to our questions. For example, heat transfer through the walls and ceiling of a typical house is never steady since the outdoor conditions such as the temperature, the speed and direction of the wind, the location of the sun, and so on, change constantly. The conditions in a typical house are not so steady either. Therefore, it is alinosl impossible to perform a heat transfer analysis of a house accurately. But then, do we really need an in-depth heat transfer analysis If the... 

SOLUTION Wo have solved this problem in Example 5-1 for the steady case, and here we repeat it for the transient case to demonstrate the application of the transient finite difference methods. Again we assume one-dimensional heat transfer in rectangular coordinates and constant thermal conductivity. The number of nodes is specified to be W = 3, and they arc chosen to be at the two surfaces of the plate and at the middle, as shown in the figure. Then the nodal spacing Ax becomes... 

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... 

You saw how the equations governing energy transfer, mass transfer, and fluid flow were similar, and examples were given for one-drmensional problems. Examples included heat conduction, both steady and transient, reaction and diffusion in a catalyst pellet, flow in pipes and between flat plates of Newtonian or non-Newtonian fluids. The last two examples illustrated an adsorption column, in one case with a linear isotherm and slow mass transfer and in the other case with a nonlinear isotherm and fast mass transfer. Specific techniques you demonstrated included parametric solutions when the solution was desired for several values of one parameter, and the use of artificial diffusion to smooth time-dependent solutions which had steep fronts and large gradients. 

In selecting metals and alloys as materials of construction, one must have knowledge of how materials fail, for example is, how they corrode, become brittle with low-temperature operation, or degrade as a result of operating at high temperatures. Corrosion, embrittlement, and other degradation mechanisms such as creep will be described in terms of their threshold values. Transient or upset operating conditions are common causes of failure. Examples include start-ups and shutdowns, loss of coolant, the formation of dew point water, and hot spots due to the formation of scale deposits on heat transfer surfaces. Identification and documentation of all anticipated upset and transient conditions are required. 

An example of the analyses that were performed and used to define the PTS licensing criteria is presented here. This example uses a vessel fabricated from rolled plate connected with axial welds to form two cylindrical shell courses. Circumferential welds connect the two shell courses. The vessel conditions used in this example are specified in Table 12.2. The pressure and temperature time histories at the vessel inner surface for the postulated transient are shown in Figs 12.2 and 12.3, respectively. The heat transfer coefficient used in the analysis was 2825W/mV°C (500 BTU/(hr-ft -°F)). Table 12.3 presents the frequency distribution for the postulated event. This event is representative of an event that is a significant contributor to TWCF. 

TWINKLE is a multidimensional spatial neutron kinetics code, whieh is patterned after steady-state codes currently used for reactor core design. The code uses an implicit finite-difference method to solve the two-group transient neutron diffusion equations in one, two, and three dimensions. The code uses six delayed neutron groups and contains a detailed multi-region fuel-clad-coolant heat transfer model for calculating point-wise Doppler and moderator feedback effects. The code handles up to 2000 spatial points and performs its own steady-state initialisation. Aside from basic cross-section data and thermal-hydraulic parameters, the code accepts as input basic driving functions, such as inlet temperature, pressure, flow, boron concentration, control rod motion, and others. Various edits are provided (for example, channel-wise power, axial offset, enthalpy, volumetric surge, point-wise power, and fuel temperatures). 

Disturbance to Feed Flow Rate. Since the flow rate of the feed at the first steady state approaches the flow rate of the second steady state, the results from dynamic simulations approach the industrial data. The reason to these deviations are, again, the misleading assumption of constant overall heat transfer coefficient during the transient state. For example, for a feed flow rate of 19 gmole/s,2 the estimated overall heat transfer coefficient is 0.012 cal/s,cm, C at steady state. However, for feed flow rate of 34.3 gmole/s (industrial data), it is around 0,015 cal/s.cm. C,... 

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