Yes, you may! I presume that the entering refrigerant is at non-zero quality and that it leaves as superheated vapor. (Advanced designs separate the liquid refrigerant from the vapor and send only the liquid to the evaporator.) In the design, you must consider it as two heat exchangers, one evaporating the refrigerant and the other, superheating it. Then for each part you will need to find the appropriate LMTD to use. The total area of the evaporator is then the sum of these areas.

Your specifications are not complete and hence only a limited answer could be given.

I agree with the previous answer. I suggest you to better explain your specifications.

The LMTD will appear only in the region of super heating only since the temperatures of both the  hot and cold fluids are constants in the phase change region. with known flow rate of refrigerant you can calculate the total enthalpy of refrigrant at evaporator inlet, and the total enthalpy at the saturated vapor state. This enthalpy difference  is the heat transferred to the refrigerant at phase change region. you need to calculate the inner and outer heat transfer coefficients  and overall heat transfer coefficient U. Knowing U , delta T, and Q you can calculate the area of evaporator in the wet region. Take care that h_i for two phase flow. Repeat for superheat region and the LMTD will have meaning in this region

@Pavan, you might see above that Alessandro and I had suggested that you give complete details so that a more satisfactory answer could be given. Presumably you have got all the clarity you needed as you did not post anything further.

I bring it up now precisely because of the answer of Mohamed Salah Hassan, who had taken your statement that  "glycol is to be maintained at -10 degree Celcius" as "glycol solution temperature is constant throughout at -10 C". As far as I can understand, you are using ethylene glycol, as the 30% solution has a freeze point of -16 C, which is quite satisfactory. Thus the glycol solution is in single phase.

If this solution is indeed in single phase, for heat removal it has to enter warmer and leave cooler even if the refrigerant temperature is constant. However, it is no more than an assumption to ease the first set of iterative calculations. Assume temperature of refrigerant as constant, make the first set of calculations for length of coil, calculate the pressure drop. With the pressure profile known throughout the length of coil, you can take proper cognizance of the VARYING refrigerant temperatures in your second iteration. You may iterate till you get satisfactory convergence of the values before and after.

Therefore, you will need to consider LMTD in both, the evaporating and the superheating sections, though separately.