Program Output

Estimated temperature rise:

This is an estimate of the increase in surface temperature based on the constant flux boundary condition solution to the heat transfer equation in one dimension.  This is overestimated by about 25% due to the approximation of a gaussian flux by a constant flux. See appendix A.2 of Adam Powell's PhD thesis at MIT (soon available on the web) for the derivation.

Peclet number:

This dimensionless number gives a rough estimate of the relative significance of convective and conductive heat transfer. If it is much less than one, conduction; if much greater than one, convection dominates. It is typically given by the ratio uL/a where u is the fluid velocity, L the length scale, and a the thermal diffusivity.

In this problem, this number compares an approximation of transient convection with transient conduction. Because convection is horizontal and conduction is vertical, we use a modified version of this parameter:

uL_z²

aL_x

where L_x and L_z are the horizontal and vertical length scales, respectively.  If we follow the derivation in appendix A.3 of Adam Powell's PhD thesis at MIT (soon available on the web), we end up with

32P(ds/dt)

pi²LR³f²sqrt(kmc_p)

(P=power, s=surface tension, T=temperature, pi=3.1415..., L=pattern length, R=spot radius, f=scan frequency, k=thermal conductivity, m=viscosity, c_p=specific heat.) If density, viscosity or surface tension coefficient is set by the user to zero, this parameter will not be calculated.

Penetration/first element:

Electron penetration range divided by first element thickness.  If small, the electron beam is for all practical purposes a surface heat source; if large, this indicates that the vertical heat distribution function G_z(z) discretizes well.  That function is given by

hP exp[-4(z/S-1/3)²]

piR²S

(h is a normalizing constant ~1.36, P is beam power, pi=3.1415..., R is beam spot radius, S is beam penetration range, z is depth below surface), which is a good fit to the curve given in Schiller's Electron Beam Technology (1980).  This function, which gives centerline heat distribution, is multiplied by G_t(t) which varies between zero and 1 according to exp(-d²/R²) (d is distance from beam centerline at time t) to give the overall generation function G(z,t).

Maximum beam heat flux:

Heat flux due to the beam at the center of the beam spot.

Total beam hit energy:

Energy per unit area imparted by one pass of the beam.

T_surf:

This is the calculated surface reference temperature of the last cycle.  This is the projection of the tangent line from the bottom of the surface layer to the surface, as described in the input section.

Evaporation rate:

The rate of evaporation of material from the surface.

Total surface heat loss:

Heat loss due to radiation and evaporation.  Values for each component are given as well.

Centerline beam flux:

This compares the real value, given by P/piR² (P is beam power, pi=3.1415..., R is beam spot radius) with the total integrated generation function used in the program, to verify the accuracy of the program's heat generation function G_z(z).

Average beam heat flux:

The "Calculated" value for this with the OneHit pattern is equal to the real centerline heat flux value (see above) times the ratio of residence time to cycle period; for the Lissidue and AdamDisk patterns it is the ratio of beam power to pattern area defined by pattern length and width (the rectangle for Lissidue and elipse for AdamDisk).  The "Actual" value is the real centerline heat flux value times the integral of the time-dependent heat generation function over one cycle.  Comparing the two allows one to verify the accuracy of the time-dependent heat generation function G_t(t) in the one hit pattern, and to determine the uniformity of the Lissidue and AdamDisk patterns.

Layer bottom temperature:

This is just the temperature at the bottom of the surface layer in the last cycle.

Heat conduction through layer bottom and surface:

This is the time integral of the slope in the bottom and top elements respectively, multiplied by the thermal conductivity.  If everything is working properly, the heat conduction through the layer bottom should be very close to the difference between actual average beam heat flux and the total surface heat loss.  If the penetration depth is much smaller than the first element thickness, the beam will act as a surface source, so the approx. heat conduction through surface will be very close to the actual average beam heat flux.

Beam heat generation:

This shows depth-dependent heat generation rate along the centerline of the beam.  At low voltage, the beam penetration range will be smaller than the first element, so for the purposes of this program the beam will act as a surface flux.  This currently shows the integral of the product of heat generation and the finite element shape function at the nodes, so for penetration significantly deeper than the first element, the surface value will appear to be half of what it should be.  This is why the units are kilowatts per centimeter squared (not cubed).  This will be changed in the future, by dividing this value at each node by the integral of the shape function.

Normalized beam history:

This shows the time-dependent ratio of beam intensity to maximum intensity at the centerline.  (See penetration/first element above.)

Bottom heat flux:

This shows the conductive flux through the bottom of the surface layer, as measured by -kDT/Dz (k is thermal conductivity, DT and Dz are temperature difference and thickness across the bottom element).  If it varies significantly (more than ~20%), increase the layer thickness or the value of bapheth or the system will be overconstrained and the surface temperature will not vary as much as it should.

Surface temperatures:

This shows the surface temperature over time.  If the number of cycles is specified as "converge", it will only show the last beam scan cycle; otherwise it will show all of them and therefore illustrate convergence of the simulation.

All temperatures:

This 3-D graph shows temperature as a function of depth and time for all cycles of the entire simulation.  It is only available when the number of cycles is specfied (e.g. not "converge").

Last cycle temperatures:

This 3-D graph shows temperature as a function of depth and time in the last cycle of the simulation (i.e. when it has converged).