Mixing tank concentration calculator

Failed validation is not billed; successful calls use the published $0.002 base price with no model tokens and no outbound network.

The constant-volume mixing model and closed-form C(t)

Assume a tank of fixed volume V is perfectly mixed, so the outlet concentration always equals the instantaneous tank concentration C(t). Salt mass Q(t) equals C(t) times V. When the flow rate is zero the exponential factor is one for every finite time, so the tank simply holds C0 unchanged. The hydraulic residence time τ equals V over r when r is positive; after a few multiples of τ the tank has largely forgotten its initial condition and sits near the inlet concentration. This capability evaluates that closed form at a single requested time; it does not integrate numerically step by step, does not handle variable volume or unequal in and out rates, and does not model incomplete mixing. Liquid enters at volumetric rate r with dissolved salt concentration c_in and leaves at the same rate r, so the volume never changes. The mass balance says the rate of change of salt equals salt in minus salt out: dQ/dt equals r times c_in minus r times C, which is r times c_in minus (r over V) times Q. That is a first-order linear ordinary differential equation with constant coefficients. Separating variables or using an integrating factor yields Q(t) equals V times c_in plus (Q0 minus V times c_in) times exp of negative r t over V. Dividing by V gives the concentration formula used here: C(t) equals c_in plus (C0 minus c_in) times exp of negative r t over V, where C0 is the initial tank concentration. As t goes to infinity the exponential vanishes and C approaches c_in, the natural equilibrium of a continuous feed of fixed strength. When C0 already equals c_in the exponential term is multiplied by zero and concentration is constant for all t.

Input contract, response fields, and validation rules

An optional precision integer from zero through fifteen sets decimal places for numeric outputs and defaults to ten. Negative volume, flow rate, concentrations, or time also fail with clear messages, because they break the physical model or the domain of the exponential. All numbers are rounded with fixed decimal places for stable goldens across runtimes. Example: volume one hundred, flow rate five, inlet concentration two, initial concentration zero, time ten yields exponential factor about zero point six zero seven and concentration about zero point seven eight seven in the same mass-per-volume unit you used for the inputs. POST a JSON body with five required finite numbers: volume (positive tank volume V), flow_rate (non-negative volumetric rate r with inflow equal to outflow), inlet_concentration (non-negative inflow salt strength c_in), initial_concentration (non-negative tank concentration C0 at t equals zero), and time (non-negative elapsed time t). Each field may arrive as a JSON number or a decimal string; aliases such as V, r, c_in, C0, and t are accepted so notebook scripts can use textbook symbols. Missing fields, empty strings, NaN, Infinity, and magnitudes beyond one times ten to the fifteen fail with invalid_input naming the field. On success the body echoes the rounded inputs, returns concentration C(t), salt_amount Q equals C times V, equilibrium_concentration equal to c_in, exponential_factor equal to exp of negative r t over V, method set to mixing_tank_first_order, the formula string, a short unit_note, and when r is positive the finite time_constant τ equals V over r. When r is zero the time_constant field is omitted so clients never see a JSON null for infinity.

Where a mixing-tank concentration API earns its keep

Learning-management systems can grade numeric free-response items by comparing student answers to this endpoint within a tolerance, rejecting negative times and zero volumes as invalid rather than silent wrong keys. Chemical and environmental engineering notebooks estimate how long a continuous stirred tank needs to approach a discharge limit when feed concentration is fixed; calling a tiny parse API avoids spinning up a full ODE solver for a single exponential.