# Set of equations

1. Feb 3, 2006

### 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

2. Feb 3, 2006

### HallsofIvy

Staff Emeritus
Are we to assume that Tm, TR, 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
$$Q_P + \rho \cdot C_P \cdot \dot V\left( {T_O - T_R } \right) = U \cdot A\left( {T_R - T_O } \right)$$
$$Q_P - \rho \cdot C_P \cdot \dot V\left( {T_R - T_O } \right) = U \cdot A\left( {T_R - T_O } \right)$$
[Tex]\left(U\cdot A+ \rho \cdot C_P \cdot \dot V\right)\left(T_R- T_O\right)= Q_P[/tex]
$$T_R- T_O=\frac{Q_P}{U\cdot A+ \rho \cdot C_P \cdot \dot V\right}$$
$$T_R= T_O+ \frac{Q_P}{U\cdot A+ \rho \cdot C_P \cdot \dot V\right}$$
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.