They could just as easily used a low cm NACA0018 or NACA2415 foil with a slatted elevator to get the same cL, but they chose a high cm foil instead.

What's a valid reason for adding slats on elevator? Or do you mean slatted tail? In either case, I can't think of a good enough reason. Slats are designed to increase the CLmax of a lifting surface; it does not increase its lift slope. Therefore, slats on tail wouldn't seem to serve much purpose. In the Zenith design, slats are likely added on the wing to allow for STOL capability. As an aside, I've also seen slats improving the stall characteristics of transonic swept wings, but apparently that's not a general case (What would be the effect of skinning over the gap in fixed leading-edge slats?).

I'm assuming using a low cm wing and a high cm horizontal tail adds to the effective moment produced by the area of the horizontal tail, allowing for a smaller horizontal tail area.

That's a good question. I did some quick math using a variation of usual airplane parameters, and here's the summary of the results:

1. The static margin is unaffected by the camber. For NACA4412, there may even be a forward shift in AC (for the same tail planform), so there may be a slight reduction in static margin.

2. Having a negatively cambered tail offloads the tail lift, exacting a small saving in trim drag (a few counts at cruise CL of 0.4). Since it offloads the tail, it may also be beneficial to preventing tail stall, but the effect is fairly small.

3. A negatively cambered tail also produces negative zero-incidence lift. The amount of negative lift can be tailored depending on the overall configuration (CG range, flaps, thrust lines) to minimize form drag.

We can derive the following trimmed lift and pitching moment equations assuming a cambered tail and zero thrust:

(1) $C_{L}=C_{L_{wb_0}}+a\alpha+\frac{S_t}{S}[C_{L_{t_0}}+a_t(i_t-\epsilon_0)]$

(2) $C_m=C_{m_{ac_{wb}}}+C_{m_\alpha}\alpha-V_H[C_{L_{t_0}}+a_t(i_t-\epsilon_0)]+(h-h_{ac_{wb}})C_{L_{wb_0}}+\frac{S_t}{S}C_{m_{ac_t}}=0$

where,

• $C_{m_\alpha}=a(h-h_{ac_{wb}})-a_t\overline{V_H}(1-\frac{\partial \epsilon}{\partial \alpha})$

• $a=a_{wb}+a_t\frac{S_t}{S}(1-\frac{\partial \epsilon}{\partial \alpha})$

• $a_t$ is the lift slope of the tail, $a_{wb}$ is the lift slope of the wingbody, $\overline{V_H}$ is the tail volume from the wingbody AC, $V_H$ is the tail volume from the CG, $\epsilon$ is the downwash on the tail

They could just as easily used a low Cm NACA0018 or NACA2415 foil with a slotted elevator to get the same CL, but they chose a high Cm foil instead.

For an elevator, we want to maximize the lift coefficient at the maximum effective control deflection, while maintaining a high degree of linearity with respect to the deflection. There is a maximum effective deflection because any higher control deflection will no longer increase the lift.

A well-designed slotted elevator is theoretically supposed to help with that. But the size of the slot, and any spoiling elements along the hinge would need to be carefully tested in the wind tunnel. Otherwise, they may end up reducing your maximum effective deflection instead. The mechanism required to achieve a good slot may also be complicated for an elevator (slotted flaps don't move like an elevator).

I'm assuming using a low Cm wing and a high Cm horizontal tail adds to the effective moment produced by the area of the horizontal tail, allowing for a smaller horizontal tail area.

That's a good question. I did some quick math using a variation of usual airplane parameters, and here's the summary of the results:

1. The static margin is unaffected by the camber. For NACA4412, there may even be a forward shift in AC (for the same tail planform), so there may be a slight reduction in static margin.

2. Having a negatively cambered tail offloads the tail lift, exacting a small saving in trim drag (a few counts at cruise CL of 0.4). Since it offloads the tail, it may also be beneficial to preventing tail stall, but the effect is fairly small.

3. A negatively cambered tail also produces negative zero-incidence lift. The amount of negative lift can be tailored depending on the overall configuration (CG range, flaps, thrust lines) to minimize form drag.

We can derive the following trimmed lift and pitching moment equations assuming a cambered tail and zero thrust:

(1) $C_{L}=C_{L_{wb_0}}+a\alpha+\frac{S_t}{S}[C_{L_{t_0}}+a_t(i_t-\epsilon_0)]$

(2) $C_m=C_{m_{ac_{wb}}}+C_{m_\alpha}\alpha-V_H[C_{L_{t_0}}+a_t(i_t-\epsilon_0)]+(h-h_{ac_{wb}})C_{L_{wb_0}}+\frac{S_t}{S}C_{m_{ac_t}}=0$

where,

• $C_{m_\alpha}=a(h-h_{ac_{wb}})-a_t\overline{V_H}(1-\frac{\partial \epsilon}{\partial \alpha})$

• $a=a_{wb}+a_t\frac{S_t}{S}(1-\frac{\partial \epsilon}{\partial \alpha})$

• $a_t$ is the lift slope of the tail, $a_{wb}$ is the lift slope of the wingbody, $\overline{V_H}$ is the tail volume from the wingbody AC, $V_H$ is the tail volume from the CG, $\epsilon$ is the downwash on the tail