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.