Tank Drain and Heat-Up Time: Method

Quasi-Steady Drain, Closed Forms, Gas Pad, Batch Heating

Scope

The first dropdown chooses one of four analyses. Drain: the time for the level of a tank to fall between two levels through an orifice, a nozzle or a discharge pipe, for six tank shapes or a pasted level table, with the gas space vented, held at a constant pressure, closed as an expanding gas pad, or sealed. Fill: the time to raise the level with a constant inflow or with a pump whose flow falls as the level rises. Equalise: two tanks of constant plan area joined by an orifice or a pipe. Heat-up or cool-down: the time for a well-mixed batch to move between two temperatures with steam or another medium in a coil or jacket, an external exchanger, an electric or fired heater, and a loss to ambient. Outside the scope: inertia of the liquid in long pipes (the flow is taken as steady at each level), two-phase or flashing discharge, vortex and air entrainment at low levels, an overflow weir (the fill simply ends at the end level), stratified or unmixed batches, boiling and evaporation from an open surface, and the calculation of the overall coefficient U from agitation and film coefficients, which is an entered value.

Conventions and units

Levels are measured up from the lowest point of the tank. Lengths, diameters, volumes, flows, pressures, densities, masses and powers go through the site's unit dictionary. Times are in seconds, with minutes or hours beside them. Temperatures are plain numbers read as °C in SI mode and °F in USCS mode; specific heats are kJ/(kg⋅K) or Btu/(lb⋅°F); overall coefficients are W/(m²⋅K) or Btu/(h⋅ft²⋅°F); viscosity is in centipoise. The unit switch does not convert these plain numbers. Pasted tables are in m, m², m³ (ft, ft², ft³) and pump curves in L/s and m (gpm and ft). Everything is computed in SI and shown in the selected system.

Tank geometry

The free-surface area A(h) and the volume V(h) are exact for each shape. Vertical cylinder and rectangular tank: A is constant. Sphere: A = π h (D − h), V = π h² (3D/2 − h)/3. Cone or frustum with the small end down: the radius grows linearly with the level, V = π h (r_b² + r_b r + r²)/3. Horizontal cylinder of length L: A = 2 L √(h (D − h)) and V = L [R² arccos((R − h)/R) − (R − h) √(2 R h − h²)]. Its two heads are treated as half-ellipsoids of depth c, which together add A = π c h (D − h)/R and V = (π c/R)(R h² − h³/3): exact for 2:1 ellipsoidal heads (c = D/4) and hemispherical heads (c = D/2). A torispherical (dished) head is entered by its depth; the ellipsoid of the same depth is close to its volume, not identical. A pasted level-area table is interpolated linearly in area; a level-volume (strapping) table is interpolated linearly in volume, so its area is constant between rows.

Drain time

While the level falls slowly compared with the speed of the outflow, the flow at each instant is the steady flow for that level, and continuity gives A(h) dh/dt = −Q(h). The time between the levels h₁ and h₂ is therefore t = ∫ A(h)/Q(h) dh. The head driving the flow is H_e = h + z + p_g/(ρ g): the level, the vertical drop z from the tank bottom to the point where the stream leaves into the atmosphere, and the gauge pressure of the gas above the liquid. For an orifice or nozzle Q = C_d a √(2 g H_e) (Torricelli's velocity with a discharge coefficient). For a pipe the velocity satisfies H_e = (1 + K + f L/d) V²/(2 g), where K sums the entrance, valve and fitting losses, the 1 is the velocity head carried away by the jet, and f is the Darcy friction factor: 64/Re in laminar flow, the Colebrook equation above Re = 2,300. The velocity is solved at every level, so the friction factor follows the falling Reynolds number.

The tool evaluates the integral numerically in level bands with a tanh-sinh rule, which handles the square-root behaviour where the head or the surface area goes to zero. Where an elementary closed form exists it is printed beside the numerical value. For a constant area, t = 2 A (√H₁ − √H₂)/(C_d a √(2g)) (Merriman, Art. 57); emptying takes exactly twice as long as passing the same volume at the constant starting flow. For a sphere, a cone and the ellipsoidal heads the area is a polynomial in the level and each term integrates to a power of H; a full sphere empties through its lowest point in 4 π D^5/2/(15 C_d a √(2g)) and a hemispherical bowl of radius r in 14 π r^5/2/(15 C_d a √(2g)) (Merriman, Art. 32). A horizontal cylinder with flat ends and the outlet at its lowest point drains in 4 L [(D − h₂)^3/2 − (D − h₁)^3/2]/(3 C_d a √(2g)); with a head offset its integral is elliptic and only the numerical value is given.

Outlet position, siphons and vortices

A positive outlet drop is a tail pipe, dip leg or hose that discharges below the tank bottom: the drop adds to the head at every level and the tank empties in a finite time even through a long pipe. A siphon over the wall behaves the same way while it stays primed, and the absolute pressure at its crest must stay above the vapor pressure of the liquid. A negative drop is an outlet in the wall above the bottom: the drain stops at the outlet. With a pipe and no head left at the end level, the laminar tail makes the time to exactly zero head unbounded, so the tool integrates to 0.1 % of the starting head and says so. Below a head of about three orifice diameters a vortex forms over a bottom outlet and the real discharge falls below the formula.

Gas space

Vented: the surface is at atmospheric pressure, provided the vent can admit air at the rate the liquid leaves. Constant pressure: a regulated blanket adds the constant head p_g/(ρ g), so the closed forms still apply with H = h + z + p_g/(ρ g). Gas pad: a closed gas volume expands as the liquid leaves, p Vⁿ = constant, from the starting absolute pressure and the gas volume at the start level (the tank's own space above the liquid plus any entered extra volume); n = 1 for a slow isothermal expansion, up to the specific-heat ratio for a fast one. Sealed: the same with the gas starting at atmospheric pressure. In the last two cases the pressure can fall below atmospheric; the flow stops at the level where the vacuum balances the liquid head (vacuum lock), which the tool finds and reports, and the shell must withstand that vacuum.

Filling and equalising

With a constant inflow the filling time is the volume to add divided by the flow. With a pump, the operating flow at each level is where the pasted pump curve (interpolated linearly) meets the system head H_static + h + h_f (Q/Q_ref)²; with bottom entry the static head rises with the level, the flow falls, and t = ∫ A(h)/Q(h) dh. If the pump reaches its shut-off head before the end level the tool reports the level where the flow stops. For two tanks of constant areas joined below both surfaces, the level difference ΔH drives the flow and d(ΔH)/dt = −Q(ΔH)(1/A₁ + 1/A₂); through an orifice t = 2 A₁ A₂ (√ΔH₀ − √ΔH)/((A₁ + A₂) C_d a √(2g)), and the common final level is the area-weighted mean of the starting levels.

Heat-up and cool-down of a batch

For a well-mixed batch of heat capacity M c (liquid plus, optionally, the vessel metal), M c dT/dt = UA_eff (T_m − T) + P − UA_L (T − T_a). The equation is linear: the batch approaches the steady temperature T_∞ = (UA_eff T_m + UA_L T_a + P)/(UA_eff + UA_L) with the time constant τ = M c/(UA_eff + UA_L), T(t) = T_∞ − (T_∞ − T₁) e^−t/τ, and t = τ ln[(T_∞ − T₁)/(T_∞ − T₂)]. A target beyond T_∞ is never reached, which is how a loss to ambient sets a ceiling on an underpowered heater. Without a loss and with steam at T_s this is the textbook t = (M c/(U A)) ln[(T_s − T₁)/(T_s − T₂)], which is the same statement as heat = U A × log-mean temperature difference × time. With constant power and no loss the rise is linear, t = M c ΔT/P. With no source the loss term alone gives Newton's law of cooling.

The effective conductance UA_eff refers the heat flow to the inlet temperature of the medium. A condensing or otherwise constant-temperature medium in a coil or jacket gives UA_eff = U A. A liquid medium of capacity rate W c_m changes temperature along the coil; with the batch at one temperature the coil has the effectiveness 1 − e^{−UA/(W c_m)}, so UA_eff = W c_m (1 − e^{−UA/(W c_m)}). With an external exchanger the batch is recirculated at the capacity rate w c: UA_eff = w c (1 − e^{−UA/(w c)}) for a constant-temperature medium and C_min ε for a liquid medium in counterflow, ε being the counterflow effectiveness. These are the batch heating and cooling forms found in the process heat-transfer texts, written with one symbol. In each case the flow sets an upper limit on UA_eff that no amount of surface can exceed.

The duty of the source is largest at the start. For steam the consumption is the heat delivered divided by the latent heat at the steam pressure (IAPWS-IF97), with the condensate leaving as saturated liquid; the starting rate sizes the control valve, the traps and the supply. The inverse options find the surface area or the heater power for a target time: the power in closed form, the area by bisection on UA_eff followed by the inverse of the arrangement's relation.

Coefficients

The discharge coefficient is an input. The U.S. Bureau of Reclamation's Water Measurement Manual gives about 0.61 for a sharp-edged orifice with full contraction; Merriman uses 0.6 for orifices with square inner edges and 0.8 where the inner edges are rounded. The overall coefficient U of a coil or jacket depends on the liquid, the agitation, fouling and the medium, and varies by more than a factor of ten between a viscous unstirred liquid and stirred water under condensing steam; use a tested or vendor value. The defaults (C_d = 0.61, U = 800 W/(m²⋅K), loss coefficient 1 W/(m²⋅K)) are illustrative.

Checks

The library is checked against Merriman's worked examples (a canal lock of 80 by 20 ft emptying through 8 ft² of orifices under 16 ft in 333 s; an orifice of 1 ft² in a 10 ft² reservoir under 9 ft passing 13.9 ft³ in the first second), every closed form against numerical quadrature on forty seeded tanks, the pipe drain, the gas pad, the pump fill and the batch equation against an independent ordinary-differential-equation solver, and Hausbrand's steam-coil example (100 kg of water from 30 to 90 °C under steam at 100 °C).