NOTES

 

rFcTo

rFcT(z)

rFcTf

Po

P(z)

The form of the local energy conservation equation which governs the microwave water heating process during stady state conditions can be written as :

dP(z)/dz = - rFcdT(z)/dz (1)

where P(z) and T(z) are the microwave power and the temperature of water at a distance z away from the inlet, r, F, and c are the volumetric mass, flow rate and specific heat of water respectively.

However the microwave power at a distance z away from the inlet is given by, [4], [7] :

P(z) = Po exp{- 2 [Integration from 0 to z] a(z') dz'} (2)

where Po and a(z') are the input microwave power and the heat transfert absorption coefficient respectively. For a WR 340 water heat transfer for which e is the thickness of the liquid slab a(z') = ß / T(z') in the temperature range 25 < T < 75° C where ß is the following physical constant :

ß = 7.35 e 230 dB.° C/m = 1.690 e dB.° C/mm = 0.194 e °C/mm where e is in mm.

Integrating eqn.(1) gives :

Po + rFcTo = P(z) + rFcT(z) = rFcTf (3)

where Po is totally absorbed along the heat transfer, To and Tf are the input and output temperatures of water respectively.

Differentiating eqn.(2) gives :

dP(z)/dz = - 2 ß P(z) / T(z) (4)

Substracting eqn.(4) from eqn.(1) yields :

rFcdT(z)/dz = 2 ß P(z) / T(z) (5)

However eqn.(3) gives :

P(z) / T(z) = rFcTf / T(z) - rFc (6)

Substituing P(z) / T(z), eq.(5) yields :

dz = dT(z) / 2 ß {Tf / T(z) - 1} (7)

Integrating eqn.(7) along the heat transfer gives :

2ßz/To = - ln (1- d) - [ln (1- d) + d] DT/To (8)

 

where DT = Tf - To and d = {T(z) - To}/DT = DT(z)/DT
DT is related to the process parameters by the formula : DT = Po / rFc

Ignoring the absorption coefficient variation with a(z') = ß / To gives the simplified equation :

2ßz*/To = - ln (1- d) (9)

 

where z* is the approximate position at ratio d.

Now let us consider a volume of water of thickness e and height h(z) described as follows :

This design is well adapted to minimize microwave reflections on the inner pipe. The corresponding heat transfer absorption coefficient a'(z') can be written as:

a'(z') = a(z') h(z')/b (10)

where b is the height of the wave guide.

By following the same procedure as for the rectangular slab, eqn.(7) becomes :

h(z)dz/b = dT(z) / 2 ß {Tf / T(z) - 1} (11)

Integrating eqn.(11) along the heat transfer gives :

V(z)/v = - ln (1- d) - [ln (1- d) + d] DT/To (12)

where V(z) is the necessary volume of water which leads to the required ratio d.

Therefore :

V(z) = [Integration from 0 to z] eh(z')dz'

By definition v is the volume of a rectangular liquid slab of eight b at a point distance To/2ß away from the inlet :

v = beTo/2ß (13)

As the thickness e is substantially smaller than the width of the heat transfer, ß varies linearly with e [7], therefore v as well as V(z) at a given ratio d do not depend upon the shape h(z) of the inner pipe.

It will be noted that the volume V(z) is the appropriate parameter for describing the microwave heating process.