Solving 2 Equations with ODE and Diff

[TSN79]

I'm trying to solve a set of two equations, one of which is an ODE. They are

$$m \cdot C_P \cdot {{dT_M } \over {dt}} = U \cdot A\left( {T_R - T_M } \right)$$

and

$$Q_P + \rho \cdot C_P \cdot \dot V\left( {T_O - T_R } \right) = U \cdot A\left( {T_R - T_O } \right)$$

I want to solve this set for $T_M$ and $T_R$, but I'm not sure about the procedure, because of the diff. Any help will be appreciated

 

[HallsofIvy]

Are we to assume that T_m, T_R, and T₀ are functions of t? If T₀ is also unknown, then you don't have enough equations. If T₀ is a known function of t, then from
$$Q_P + \rho \cdot C_P \cdot \dot V\left( {T_O - T_R } \right) = U \cdot A\left( {T_R - T_O } \right)$$
$$Q_P - \rho \cdot C_P \cdot \dot V\left( {T_R - T_O } \right) = U \cdot A\left( {T_R - T_O } \right)$$
$$\left(U\cdot A+ \rho \cdot C_P \cdot \dot V\right)\left(T_R- T_O\right)= Q_P$$
$$T_R- T_O=\frac{Q_P}{U\cdot A+ \rho \cdot C_P \cdot \dot V\right}$$
$$T_R= T_O+ \frac{Q_P}{U\cdot A+ \rho \cdot C_P \cdot \dot V\right}$$
Now put that function into
$$m \cdot C_P \cdot {{dT_M } \over {dt}} = U \cdot A\left( {T_R - T_M } \right)$$
and solve the differential equation.

 

[tacman]

To solve this set of equations, you will need to use a method called simultaneous differential equations. This involves solving both equations at the same time, rather than separately. The first step is to rearrange the equations to isolate the variables you want to solve for, in this case T_M and T_R. For the first equation, you can divide both sides by mC_P to get:

{{dT_M} \over {dt}} = {U \cdot A \over mC_P} \cdot (T_R - T_M)

Similarly, for the second equation, divide both sides by \rho C_P \cdot \dot V to get:

{T_O - T_R \over \dot V} = {U \cdot A \over \rho C_P \cdot \dot V} \cdot (T_R - T_O)

Now, you can substitute these expressions for dT_M/dt and (T_O - T_R)/\dot V into the first equation, and solve for T_R. Then, substitute this value for T_R into the second equation and solve for T_M. This will give you the values for both T_M and T_R at any given time.

However, keep in mind that this method may not always give an exact solution, and you may need to use numerical methods to approximate the values. It's also important to check your units and make sure they are consistent throughout the equations.

I hope this helps and good luck with solving your set of equations!