Edit to add: The OP states the applicaiton is a heat pipe, so this answer assumed the yellow fluid is a liquid and moves by free convection. It turns out the yellow fluid will undergo a phase change, which changes a lot.

This is an outline for a solution that should get you going:

Step one - find an equation describing the convection of the yellow medium.

$dot{m}_{yellow}$ is determined by $T_{in}$, $T_{out}$ (340°C) and the shape and physical properties of the heat exchanger, how they are situated to each other, etc. To define this function $dot{m}_{yellow}(T_{in},T_{out})$ is the first, and likely most difficult, step.

Step two - find your $dot{m}_{yellow}$

Find a value that satisfies $$dot{m}_{yellow}(T_{in},T_{out}) * c_{p, yellow} * (T_{out} - T_{in}) = Q_{red}$$

If your function for $dot{m}_{yellow}$ is a line (IMO unlikely) there will be many possible values, else it should be only one.

Size heat exchangers

for the $dot{m}_{yellow}$ value and the other values found above.

Simulate the resulting system

with the fixed inlet temperatures for the blue and red fluids and some arbitrary starting temperatures, to see if the system indeed approaches the values you assumed. Here the NTU method is probably the best approach.

Background

Total energy lost from fluid

The heat transfer rate $dot{Q} $ if you know for a pipe with fluid $f$ (where f: Red, Yellow, Blue) the temperature at input and output, the mass rate, and the heat capacity of the material is given by:

$$dot{Q}_f = dot{m}_fcdot C_{p,f}(T_{f,o}- T_{f,i}) $$

So for the Red-yellow pipe you have:

$$dot{Q}_{ry} = -dot{m}_rcdot C_{p,r}(T_{r,o}- T_{r,i}) = dot{m}_ycdot C_{p,y}(T_{y,o1}- T_{y,i1}) $$ $$dot{Q}_{ry} = -dot{m}_rcdot C_{p,r}(130-540) = dot{m}_ycdot C_{p,y}(340- T_{y,i1}) $$ $$dot{Q}_{ry} = dot{m}_rcdot C_{p,r}(410) = dot{m}_ycdot C_{p,y}(340- T_{y,i1}) $$

So for the Blue-yellow pipe you have:

$$dot{Q}_{by} = dot{m}_bcdot C_{p,b}(T_{b,o}- T_{b,i2}) = -dot{m}_ycdot C_{p,y}(T_{y,o2}- T_{y,i2}) $$ $$dot{Q}_{by} = dot{m}_bcdot C_{p,b}(300- 80) = -dot{m}_ycdot C_{p,y}(T_{y,o2} - 340 ) $$ $$dot{Q}_{by} = dot{m}_bcdot C_{p,b}(220) = -dot{m}_ycdot C_{p,y}(T_{y,o2} - 340 ) $$

Exchange due to conductive heat transfer

At that point you need the logarithmic mean temperature difference $Delta T_{lm}$. Essentially, it estimates an equivalent temperature that you can use for calculating the heat transfer between and exchange surface A, with a coefficient of conductivity of k. Then the heat transfer rate is :

$$dot{Q} = kAcdotDelta T_{lm} $$

Where:

$$Delta T_{lm} = frac{Delta T_1-Delta T_2}{ln (Delta T_1/Delta T_2)}$$

The $Delta T_{lm}$ has different definitions of $Delta T_1$ and $Delta T_2$ for parallel and counterflow exchangers.

• for counterflow (Red-yellow)
  • $Delta T_1 = T_{r,i}-T_{y,o1} = 540 -340 = 200$ : temperature difference at one exit (at the center of the drawing)
  • $Delta T_2 = T_{r,o}-T_{y,i1} = 340 -T_{y,i1} $ : temperature difference at other Exit (bottom of the drawing).

So: $$dot{Q}_{ry} = k_{ry}A_{ry} frac{200-(340 -T_{y,i1} )}{ln frac{200}{340 -T_{y,i1}}} = k_{ry}A_{ry} frac{-140 +T_{y,i1} }{ln frac{200}{340 -T_{y,i1}}} $$

• for parallel flow (Blue-yellow)
  • $Delta T_1 = T_{y,i2}-T_{b,i} = 340 - 80 =260$ : temperature difference at one exit (at the center of the drawing)
  • $Delta T_2 = T_{y,o2}-T_{b,o} = T_{y,o2} - 300$ : temperature difference at other Exit (bottom of the drawing).

So: $$dot{Q}_{by} = k_{by}A_{by} frac{260-(T_{y,o2} - 300 )}{ln frac{260}{T_{y,o2} - 300}} = k_{by}A_{by} frac{560 -T_{y,o2} }{ln frac{260}{T_{y,o2} - 300}} $$

Equality between change in heat capacity and conductivity transfer rate

At this point it is useful to remind that the heat transfer rate due to change in heat capacity in the red fluid is equal to the heat transmitted conductively between the red-yellow interface. Therefore you can write:

• for the red-yellow interface:

$$dot{Q}_{ry} = dot{m}_rcdot C_{p,r}(410) = k_{ry}A_{ry} frac{-140 +T_{y,i1} }{ln frac{200}{340 -T_{y,i1}}} $$

or simply (in $color{red}{text{red}}$ color I am highlighting unknown quantities:

$$dot{m}_rcdot C_{p,r}(410) = k_{ry}color{red}{A_{ry}} frac{-140 +color{red}{T_{y,i1}} }{ln frac{200}{340 -color{red}{T_{y,i1}}}} $$

• Similarly for the blue-yellow interface, ultimately you get:

$$dot{m}_bcdot C_{p,b}(220) = k_{by}color{red}{A_{by}}frac{560 -color{red}{T_{y,o2}} }{ln frac{260}{color{red}{T_{y,o2}} - 300}} $$

As you can see you have seemingly two equations with four unknowns. However there still a dependence from the equations right at the top:

$$dot{m}_rcdot C_{p,r}(410) = color{green}{dot{m}_y}cdot C_{p,y}(340- color{red}{T_{y,i}}) $$ $$dot{m}_bcdot C_{p,b}(220) = -color{green}{dot{m}_y}cdot C_{p,y}(color{red}{T_{y,o2}} - 340 ) $$

So essentially you have 1 degree of freedom (3 equation and 4 unkwowns).

How to solve

As I said just above you have 3 equation and 4 unkwowns. To sum them up:

$$begin{cases} dot{m}_rcdot C_{p,r}(410) = k_{ry}color{red}{A_{ry}} frac{-140 +color{red}{T_{y,i1}} }{ln frac{200}{340 -color{red}{T_{y,i1}}}} dot{m}_bcdot C_{p,b}(220) = k_{by}color{red}{A_{by}}frac{560 -color{red}{T_{y,o2}} }{ln frac{260}{color{red}{T_{y,o2}} - 300}} dot{m}_rcdot C_{p,r}(410) = color{green}{dot{m}_y}cdot C_{p,y}(340- color{red}{T_{y,i1}}) dot{m}_bcdot C_{p,b}(220) = -color{green}{dot{m}_y}cdot C_{p,y}(color{red}{T_{y,o2}} - 340 ) end{cases}$$

assuming you know:

• $dot{m}_r$ the mass rate of the red fluid
• $C_{p,r},C_{p,y},C_{p,b}$: the heat capacities
• $k_{ry},k_{by}$: the coefficient of heat conductivity

Then one possible way is the following:

1. You set a temperature for the input for the yellow material $T_{y,i1}$. Then you can find:
2. You can now find $A_{ry}$ from the 1st equation and assuming you know the contact geometry the required length $L_{ry}$ for the red yellow interface.
3. you can find $m_{y}$ from the 3rd equation
4. you can find $T_{y,o2}$ you can use the 4th equation, since you just calculated $m_{y}$.
5. you can find $A_{ry}$ from the 1st equation and assuming you know the contact geometry the required length $L_{by}$ for the blue yellow interface.
   $$ kAcdotfrac{Delta T_1-Delta T_2}{ln (Delta T_1/Delta T_2)}= dot{Q} = m_scdot C_{p,s}(T_{s,o}-T_{s,i}) $$

Similar problem (only a single counterflow). A similar problem can be found at this link