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of order higher than 1 in the time dimension often run into serious numericalstability problems Schiesser and Silebi successfully solved the advection equation [6]
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ut = Vux
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and the related partial differential equation
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which models a simple heat exchanger with flow velocity V (Section 7-14) [7] But direct application of the space-derivative submodels in Table 7-1 to the wave equation
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fails Schiesser [6] solved this problem by deriving a second-derivative operator for computing uxx in one step, but boundary-condition setting becomes more complicated For problems with more than one space dimension, the vector compiler produces ordinary differential equations just as easily, for one can treat, for example, two-dimensional arrays as equivalent vectors (Section 3-11) But assignment of multidimensional boundary values is likely to be cumbersome, just as it would be in a Fortran program An intelligent choice of the coordinate system can simplify model and program, as in the example of Figure 7-12
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7-14 A Simple Heat-exchanger Model In the simple heat-exchanger model programmed in Figure 7-9a, u(x, t) is the temperature of a fluid running through a tube between x = 0 and x = L The initial fluid temperature is u = T0c = 0 The fluid transfers heat to or from a surrounding constant-temperature annulus; longitudinal heat conduction is neglected An initial step change Tec in the inlet temperature at x = 0 then travels down the tube by advection u(x, t) satisfies the partial differential equation
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ut = Vux + a(U u)
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where V is the constant fluid velocity, and U the annulus temperature, which is assumed to be constant here The program in Figure 7-9 solves this partial differential equation using the second-order Schiesser differentiation operator as in Section 7-11
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-- SIMPLE HEAT EXCHANGER (Schiesser and Silebi, 1997) --------------------------------------------------------------------------------------------display N1 | display C8 | display R --------------------------------------------------------------------------------------------- -Schiesser numerical-differentiation operator -ARRAY vx[1], v[1] | -dummy arrays for submodel SUBMODEL DDx(n$, bb$, v, vx) Vector vx = (v{1} - v{-1}) * bb$ vx[1] = (-3 * v[1] + 4 * v[2] - v[3]) * bb$ vx[n$] = (3 * v[n$] 4 * v[n$ - 1] + v[n$ - 2]) * bb$ end --------------------------------------------------------------------------------------------L = 100 | -heat-exchanger tube length V = 10 | -flow velocity rho = 1 | -density of fluid in tube CP = 1 | -specific heat of fluid D=2 | -tube diameter H = 01 | -heat-transfer coefficient a = 4 * H/(rho * CP * D) -Tac = 100 | -constant annulus temperature T0c = 0 | -initial temperature in tube Tec = 50 | -tube-entry temperature -------------------------------------------------------------------------------------------------------------n = 201 | STATE u[n] | ARRAY ux[n] -DX = L/(n-1) | bb = 05/DX DT = 00005 | TMAX = 10 | NN = 20000 | scale = 100 irule 4 | RK4 rule ---------------X = 05 * (n-1) * DX/V | -time delay used in theoretical solution ---------------------------------------------------------------------------------------------program initial conditions u[k] for k = 1 to n | u[k] = T0c | next U = Tac | constant annulus temperature drun | write n = ;n -------------------------------------------------------------------------------------------------------------DYNAMIC -------------------------------------------------------------------------------------------------------------u[1] = Tec | set tube-entry boundary value for u invoke DDx(n, bb, u, ux) | -differentiate u to get ux Vectr d/dt u = - V * ux + a * (U - u) ------------------------------compare u with theoretical solution f f = 2 * (Tac + (T0c Tac * exp(-a * t) * (1 - swtch(t - X)) + swtch(t - X) * ((Tec - Tac) * exp(-a*X))) - scale uu = 2 * u[05*(n-1)] scale | errx5 = 25 * (uu - f) - 05 * scale dispt uu, errx5, f
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FIGURE 7-9a Program for computer simulation of a simple heat exchanger
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n = 201
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FIGURE 7-9b Theoretical and computed time histories of the heat-exchanger tube temperature u(x, t) for x = L/2
For each value of x along the tube, the theoretical solution in Figure 7-9 combines an exponential function increasing from u(x, t) = 0 to u(x, t) = U with a step-function temperature change due to advection of the inlet temperature step Figure 7-9b compares this theoretical solution with the numerical solution The oscillatory response shown for n = 201 and 501