Nodal temperatures

To carry out a numerical solution, a single strip of quadrilateral elements is placed along the x-axis, and all nodal temperatures are set Initially to zero. The right-hand boundary is then subjected to a step Increase in temperature (T(H,t) - 1.0), and we seek to compute the transient temperature variation T(x,t). The flow code accomplishes this by means of an unconditionally stable time-stepping algorithm derived from "theta" finite differences a solution of ten time steps required 22 seconds on a PC/AT-compatible microcomputer operating at 6 MHz. 

We thus have nine equations and nine unknown nodal temperatures. We shall discuss solution techniques shortly, but for now we just list the answers ... 

The residuals are relaxed to zero by changing the assumptions of the nodal temperatures. The largest residuals are usually relaxed first. 

As each nodal temperature is changed, a new residual must be calculated for connecting nodes. 

Next, the new values of the nodal temperatures 7, are calculated according to Eq. (3-32), always using the most recent values of the Tj. 

Choose progressively smaller values of Ax and observe the behavior of the solution. If the problem has been correctly formulated and solved, the nodal temperatures should converge as Ax becomes smaller. It should be noted that computational round-off errors increase with an increase in the number of nodes because of the increased number of machine calculations. This is why one needs to observe the convergence of the solution. 

Apply the Gauss-Seidel technique to obtain the nodal temperatures for the four nodes in Fig. 3-6. 

One may use a variable mesh size in a problem with a finer mesh to help in regions of large temperature gradients. This is illustrated in the accompanying figure, in which Fig. 3-6 is redrawn with a fine mesh in the corner. The boundary temperatures are the same as in Fig. 3-6. We wish to calculate the nodal temperatures and compare with the previous solution. Note the symmetry of the problem T, = T2, Ty =T4, etc. 

A liner of stainless steel (k = 20 W/m °C), having a thickness of 3 mm, is placed on the inside surface of the solid in Problem 3-55. Assuming now that the inside surface of the stainless steel is at 500°C, calculate new values for the nodal temperatures in the low-conductivity material. Set up your nodes in the stainless steel as necessary. 

The two-dimensional solid shown in the accompanying figure generates heat internally at the rate of 90 MW/m2. Using the numerical method calculate the steady state nodal temperatures for k = 20 W/m °C. 

The equations above have been developed on the basis of a forward-difference technique in that the temperature of a node at a future time increment is expressed in terms of the surrounding nodal temperatures at the beginning of the time increment. The expressions are called explicit formulations because it is possible to write the nodal temperatures T g.V explicitly in terms of the previous nodal temperatures Tpm In this formulation, the calculation proceeds directly from one time increment to the next until the temperature distribution is calculated at the desired final state. 

We also have p = 7817 kg/mJ and c = 460 J/kg °C, and use the thermal resistance-capacitance formula assuming that the resistances are evaluated at the arithmetic mean of their connecting nodal temperatures i.e., R2-4 is evaluated at (T3 + T4)/2. 

The fin in Prob. 3-49 is initially uniform in temperature at 300°C and then suddenly exposed to the convection environment. Select an appropriate Ar and calculate the nodal temperatures after 10 time increments. Take p = 2200 kg/m3 and c = 820 J/kg °C. 

We number the nodes 0, 1, and 2. The temperature at node 0 is given to be Tq = O C, and the temperatures at nodes 1 and 2 are to be determined. This problem involves only two unknown nodal temperatures, and thus we rieed to have only two equations to determine them uniquely. These equations are obtained by applying the finite difference method to nodes 1 and 2,... 

SOLUTION A long triangular fm attached to a surface is considered. The nodal temperatures, the rate of heat transfer, and the fin efficiency are to be determined numerically using six equally spaced nodes Assumptions 1 Heat transfer is steady since there is no indication of any change v ilh time. 2 The temperature along the fm varies in Ihe x direction only, 3 Thermal conductivity is constant. 4 Radiation heat transfer is negligible. 

TliCf riiiite difl erence fomiulatioii of steady heal conduction problems usu ally results in a system of iV algebraic equations in /V unknown nodal temperatures that need to be solved siiiiullaneously. When Af is small (such as 2 or 3), we can use the elementary elimination method to eliminate ail unknowns except one and then solve for that unknown (sec Example 5-1). The other unknowns are then determined by back substitution. When W is large, which is usually Uie case, the elimination luelliod is not practical and we need to use a more systematic approach that can be adapted to computers. 

SOLUTION Heat transfer in a long L-shaped solid bar with specified boundary conditions is considered. The nine unknown nodal temperatures are to be determined with the finite difference method. 

This completes the development of finite difference formulation for this problem. Substituting the given quantities, the system of nine equations for the determination of nine unknown nodal temperatures becomes... 

This problem involves radiation, which requires the use of absolute temperature, and thus all temperatures should be expressed in Kelvin. Alternately, we could use °C for all temperatures provided that the four temperatures in the radiation terms are expressed in the form (T + 273) Substituting the given quantities, the system of nine equations for the determination of nine unknown nodal temperatures in a form suitable for use with an iteration method becomes... 

The nodal temperatures in transient problems normally change during each time step, and you may be wondering whether to use temperatures at the previous time step i or the ir u time step i + 1 for the terms on the left side of Eq. 5-39. Well, both arc reasonable approaches and both are used in practice. The finite difference approach is called the explicit method in the first case and the iinplicit method in the second case, and they are expressed in the general form as (Hg. 5-39)... 

The explicit and implicit methods have their advantages and disadvantages, and one method is not necessarily better Ilian the other one. Next you will see that the explicit method is easy to implement but imposes a limit on the allowable time step to avoid instabilities in the solution, and the iinplicit method requires the nodal temperatures to be solved simultaneously for each time step but imposes no limit on the magnitude of the time step. We limit the discussion to. bne- and two-dimensional cases to keep the complexities at a manageable leyel, but the analysis can feadily be extended to threc-dimen.sional ca.ses and other coordinate systems. 

Once tlte foimulatton (explicit or implicit) is complete and the Initial condition is specified, the solution of a transient problem is obtained by marching in time using a step size of At as follows select a suitable time step At and determine the nodal temperatures from the initial.condition. Taking the initial temperatures as the previous solution T, , at f = 0, obtain the new solution Tjf at all nodes at time t = At using the transient finite difference relations. Now using the solution just obtained at r = At as the previous solution Tj obtain the new solution at f = 2At using the same relations. Repeat the process until the solution at the desired time is obtained. 

The explicit method is easy to use, but it suffers from an undesirable feature that Severely restricts its utility the explicit method is not unconditionally stable, and the largest permissible value of the lime step At is limited by the stability criterion. If the time step At is not sufficiently small, the solutions obtained by the explicit method may oscillate wildly and diverge from the actual solution. To avoid such divergent oscillations in nodal temperatures, the value of Af must be maintained below a certain upper limit established by the stability criterion. It can be shown mathematically or by a physical argument ba.sed pfl thc second law of thermodynamics tliat the stability criterion is satisfied if the coefficients of alt in the Tjj, expressions fcalled the primary... 

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