Solving 1 plate problem assuming finite conduction between hot and cold side.

1_plate.py

from sympy import *
init_session(quiet=True)
from sympy.abc import epsilon, sigma, c, w
var('P1 P2 P12 Pin T1 T2')
const = dict(sigma=5.67e-8,  # Stefan-Boltzmann constant
             epsilon=1.0, # emissivity of flat black paint
             c = 401.)  # W/mK, conductivity of copper plate


# In[111]:


hot = Eq(P1, Pin - P12); hot


# In[112]:


cold = Eq(P2, P12); cold


# In[113]:


p12 = Eq(P12, c * w * (T1 - T2)); p12


# In[114]:


p1 = Eq(P1, epsilon * sigma * T1**4)
p2 = Eq(P2, epsilon * sigma * T2**4); p2


# In[115]:


hot = hot.subs({P1: p1.args[1], P12: p12.args[1]}); hot
cold = cold.subs({P2: p2.args[1], P12: p12.args[1]}); cold


# In[116]:


vals = dict(Pin = 400,  # W, input power on hot side
            w = 0.015)  # m, plate width
hotv = hot.subs({**const, **vals})
coldv = cold.subs({**const, **vals}); coldv


# In[117]:


(t1, t2) = nonlinsolve([hotv, coldv], (T1, T2)).args[1]
print(f'T1 = {t1:.2f}K\nT2 = {t2:.2f}K\nTavg = {(t1+t2)*.5:.2f}K')
print(f'Texp = {((200/epsilon/sigma)**.25).subs({**const}):.2f}K')


# In[118]:


p1v = p1.subs({**const, T1: t1}).args[1]; print(f'P1 = {p1v:.2f}W')
p2v = p2.subs({**const, T2: t2}).args[1]; print(f'P2 = {p2v:.2f}W')
print(f'Pavg = {(p1v+p2v)*.5:.2f}W')
print(f'P12 = {p12.subs({**const, **vals, T1: t1, T2: t2}).args[1]:.2f}W')