Adopted model description

Figure 3.8: Representation of the dual channel model [9]

where T is the gas temperature, Te is the average ionization temperature and α is the proportionality constant. The electrical conductivity value varies along the radial direction, in particular it is maximum at the center of the arc, for r = 0, and zero outside when r = Ra. By calculating an average value:

¯σ = 1 2σM AX

the current density in the central region will be:

j = ¯σE

The electrical power deposited in the fluid is converted into thermal energy and conducted radially towards the edge of the arc. From the energy balance per unit of length it follows:

1

2σ_{M AX}E²πR_a² = 2πRa

A

k∂T

∂r

B

r=Ra

The temperature gradient can be approximated as below:

−∂T

∂r ≈2Tc− Te

R_a

where Tcindicates the temperature on the axis of the constrictor and Teindicates the temperature on the edge of the arc; replacing the previous equation gives an expression for the electric field E as a function of the radius of the electric arc Ra:

E = 2

ó

2k(Tc− T_e) Rs2

σM AX

or by substituting the expression of electrical conductivity in the form previously assumed:

E = 2 R_a

ó2k α

Now, the total current inside the electric arc is defined by the current density which multiplies the area of the arc itself. Taking into account the variation of the arc radius along the axial direction of the thruster, we can express the electric current with the following integral:

I =^{Ú Ra}

0 2πrσE dr

Using once again the average value for the electrical conductivity and replacing the expression obtained for the electric field E the dependence on the radius Ra and the temperature Tc at the center of the electric arc is also obtained for the current I:

I = 1

2σ_{M AX}πR_a²E or

I = πRa(Tc− Te)√ 2kα

The heat flow per unit area is transmitted from the arc column region to the surrounding flow. The balance of the energy exchanged at the edge of the electric arc is written below:

2kT_e− T_c R_a = kg

T_e− T_w R − R_a where

T_w is the wall temperature and kg is the thermal conductivity of the gas surrounding column arc. From the latter another form is obtained to express the difference between the temperatures at the center and at the edge of the arc. This form, substituted in the expression of the current, leads to:

I = πRa

√2αkk_g 2k

R_a

R − R_a(Te− T_w)

It should be noted that the Ra that varies along the axial direction is not known a priori, while the value of the radius of the constrictor R will be assigned.

Multiplying the expressions obtained by the electric field and the current, a new form can be obtained to express the power per unit of length dissipated by the electric arc:

EI = 4πk (Tc− Te)

This power is deposited in the propellant gas. Part of this gas reaches the temperature Te and contributes to the growth of the electric arc. If we consider an infinitesimal length dx the power absorbed by the mass flow is:

4πk (Tc− T_e) = cp(ρu)_e2πRa

dR_a dx dx

where (ρu)_e indicates the mass flow at the edge of the arc and it must be estimated. So, under the assumption of 1D and non-viscous flow, the Bernoulli equation is valid and therefore in our case:

p+ ρu² = const

but if the pressure is independent of the radial direction at a given axial station then it is also legitimate to consider radius-indipendent ρu². Furthermore, using the equation of state of ideal gases, the mass flow can be written as follows:

ρu=^ñρ(ρu²) =

óp(ρu²) RT Thus, we have:

(ρu)_e (ρu)_out ≃

óT_out T_e where

(ρu)_out = ˙m π¹R²− R_a²² and therefore

(ρu)_e = ˙m π¹R²− R_a²²

óTout

T_e

Now we can replace the latter in the power balance and we have:

˙m π¹R²− Ra22

óT_out

T_e R_adR_a

dx c_p(Te− T_out) ≃ 2k (Tc− T_e)

or by making the dependence on the current and on the arc radius appear in the second member and reworking the expression:

R_a² R²− R_a²

dR_a

dx ≃ I^ñ2k_α^ñ_T^Te

out

˙mcp(Te− T_out)

To simplify calculations, we refer to non dimension quantities which we will indicate with an asterisk apex and which will be defined starting from the reference ones calculated from a priori known quantities.

For the reference current value we assume:

I_ref = πR√

2αk (Te− T_w) so

I^⋆ = I I_ref

Relating the radius of the electric arc to the radius of the constrictor taken as a reference dimension, the non dimension quantity is defined:

r_a= R_a R

Furthermore, assuming that the thermal conductivity of the outer flow region is about half of the relative conductivity to the inner region, it is defined:

λ = k_g 2k ≃ 1

4

Now, it is possible to find the link between the non dimension quantities I^⋆ and r_a with fixed λ:

I^⋆ = λ r_a² 1 − ra

The resolution of the second degree equation with respect to ra leads to:

r_a²− I^⋆

λr_a+ I^⋆ λ = 0 r_a = 2

1 +^ñ1 + ^4λ_I^⋆

From the latter it can be seen that by increasing the current the value of Ra

tends to one and this means that the electric arc occupies the entire passage section

of the constrictor.

With a similar reasoning, an expression for the electric field can be found. We define Eref and therefore E^⋆ is obtained:

Eref = 2 R

ó2k α

E^⋆ = E E_ref or

E^⋆ = 1

r_a = 1 +^ñ1 + ^4λ_I^⋆ 2

and from this you can see the typical behavior of the electric discharge or the negative voltage-current characteristic, in fact, as the current increases, the electric field and therefore the voltage decrease.

We also introduce:

x_ref = 1 2π

˙mcp

k

óT_out T_e thus

x^⋆ = x xref

The power balance is written in non-dimension form:

r_a² 1 − ra2

dr_a dx^⋆ =

AT_e− T_w T_e− T_out

ó T_w T_out

B

I^⋆

Since Teis much larger than both Tw and Toutand the latter are of the same order of magnitude, the quantity in brackets to the second member can be approximated to one.

r_a² 1 − ra2

dr_a dx^⋆ ≃ I^⋆

Its integration allows to evaluate the Ra to R ratio as a function of x^⋆or in other words how the electric arc develops in the radial direction along the length of the constrictor. We denote the radius of the electric arc with Ra,L in the exit section of the constrictor where the flow reaches the sonic condition. The mass flow rate will be:

˙m = Γptotπ¹R²− R²_a,L²

√RT_tot

Clearly R is the universal gas constant while R is radius of constrictor, Γ is the function of Mach which for Mach equal to one is:

Γ =√ γ

A 2

γ+ 1

B_2(γ−1)^γ+1

Also for the mass flow rate we introduce a reference value with respect to which we define the non dimension quantity:

˙mref = ptotπR²

√RTtot

Γ and

˙m^⋆ = ˙m

˙mref

carrying out the appropriate steps leads to:

˙m^⋆ = 1 − ra,L2

with

r_a,L = Ra,L

R The voltage accumulated in the constrictor is:

V_constr =^{Ú L}

0

E dx= Erefx_ref

Ú L^⋆ 0

E^⋆dx^⋆ If we define:

V^⋆_constr= V_constr Vref

where we assume the denominator equal to the product Erefx_ref. Remembering that:

E^⋆ = 1/ra

dr_a

dx^⋆ = (1 − ra2) r_a² I^⋆

changing the integration variable we obtain:

V^⋆_constr=^{Ú ra,L}

0

r_a

(1 − ra2) I^⋆dr_a and solving:

V^⋆_constr= − 1

2I^⋆ ln¹1 − ra,L22

It assumes that the arc growth is simply due to the flow expansion and, although attachment to the anode is a difficult problem to deal with, it is experimentally observed that the attachment point is downstream of the constrictor exit. On the basis of this we try to estimate the voltage drop in this region where the flow is supersonic. Looking at the Fig. 3.9 downstream of the constrictor, we see that:

Figure 3.9: Representation of the electric arc growth [9]

Ra ≃ Ra,L

R_N R_c dx= dR_N

tan θ and remembering

E = 2 R_a

ó2k α

we obtain:

∆Vnozzle ≃2

ó2k α

1

r_a,Ltan θln³R_N,att R_c

4

It is also necessary to consider the voltage drop at the anode and cathode, the causes of which have already been discussed and empirical values will be assumed for these quantities.

Nel documento Preliminary design of an arcjet in the 1kW class for space application (pagine 47-55)