Adiabatic mixer energy balance

Two streams mix adiabatically; the outlet temperature is the enthalpy-weighted average of the inlets — the linear building block of every energy balance.

Governing equation

$T_{\mathrm{out}}$

Mixed-outlet temperature (K) — the unknown.

$\dot n_1,\,\dot n_2$

Inlet molar flows (mol s⁻¹).

$T_1,\,T_2$

Inlet temperatures (K).

$c_p$

Molar heat capacity (J mol⁻¹ K⁻¹), taken constant.

Derivation

1. 1
   $$\sum_{\mathrm{in}} \dot n_j h_j(T_j) = \dot n_{\mathrm{out}} h(T_{\mathrm{out}})$$

   Steady-state energy balance on an adiabatic mixer: enthalpy in equals enthalpy out.

2. 2
   $$h(T) - h(T_{\mathrm{ref}}) = c_p (T - T_{\mathrm{ref}})$$

   With a constant heat capacity the enthalpy is linear in temperature about a reference.

3. 3
   $$\dot n_1 c_p (T_1 - T_{\mathrm{out}}) + \dot n_2 c_p (T_2 - T_{\mathrm{out}}) = 0$$

   Substituting and cancelling the reference gives the balance; T_out is the flow-weighted mean inlet temperature.

Jacobian

The matrix $J_{ij} = \partial F_i / \partial x_j$ of the residual vector with respect to the unknowns, at the representative operating point. Every value below is computed by the exact forward-mode autodiff engine the solver uses — click a non-zero cell to see the symbolic derivative, each one verified against that engine in our test suite. This is the block a Newton step factorizes.

References

• Felder & Rousseau, Elementary Principles of Chemical Processes, 3rd ed., Ch. 8.