Technical calculations

d) Example:

 

Evaporator feed rate: EFR = 50000 kg/h

Total solids of evaporator feed: EFS = 12.2 %

Total solids of the concentrate: EPS = 48.0 %

Total solids of powder from spray dryer: DPS = 94.2 %

Total solids of powder from fluid bed: BPS = 97.2 %

 

There are examples of calculations in all the following sections and the results are rounded up to the reasonable decimals. However, the calculations have been made using the exact figures.

If an amount Aa of atmospheric air with humidity ya is to be heated from a temperature t1 to a temperature t2, the amount of heat Q necessary is calculated by the equation:

If A_a =30000 kg/h, y_a =0.009 kg/kg, t₁=15°C and t₂=200°C then from equation [3,8] ca₁=0.240, ca₂=0.245 and from equation [3,10] cv₁=0.445 and cv₂=0.463 kcal/kg/°C. The result is:

Two quantities of air A₁ and A₂, with humidities y₁ and y₂ and temperatures t₁ and t₂ are to be mixed, the temperature t₃ of the mixed air calculated by a simplified method is as follows:

Example: 

A₁ = 50000 kg/h, A₂ = 10000 kg/h, t₁ = 90°C, t₂ = 20°C,

y₁ = 0.0442 kg/kg and y₂ = 0.007 kg/kg:

For a more precise calculation, heat capacities of air must be considered. From equation [3,8] values are ca₁ = 0.240 kcal/kg/°C at 90°C and ca₂ = 0.241 kcal/kg/°C at 20°C and calculation of the resulting air humidity y₃ and enthalpy h₃ is by:

The enthalpy of both components using heat capacities of water vapour from equation [3,10] at 20°C cv₁ = 0.445,at 90°C cv₂ = 0.451 and using equation [3,11]:

The values ca₃ and cv₃ are calculated from equation [3,8] and equation [3,10] using the approximate mixing temperature calculated by [6,19], i.e. 78°C which are 0.241 and 0.450 kcal/ kg/°C respectively.

Using figures from the previous example and equations [3,5] through [3,7] and [3,13], the dry air rates Ad_n, water vapour rates Av_n and air densities ρ_n are:

Thus:

 

V1 = A1/_ρ1 = 50000/0.9484 = 52722 m3/h

V2 = A2/_ρ2 = 10000/1.1997 = 8336 m3/h

V3 = A3/_ρ3 = 60000/0.9804 = 61199 m3/h

Using a Pitot tube (Fig.6.1.) it is possible to measure the static pressure P_s, dynamic pressure P_d and total pressure P_t.

where: 

P_d = dynamic pressure in mm water gauge

g = gravity constant 9.81 m/s²

ρ = density of air kg/m³

v = air velocity in m/s.

In practice, it is not easy to measure exactly the amount of air passing through a duct, filter or spray dryer. The methods available are listed below:

 

a) Measuring the air velocity and duct area.

 

If the air velocity is measured by Pitot tube or by wind or hot wire anemometer in a duct of SA

area, then the volume of air flow V in m³/h is:

Knowing the air temperature t and humidity y, the air rate A in kg/h can be calculated using equations [3,12], [3,13] and [6,23]. In a similar way, for round ducts of diameter D, the equation is:

where: 

A = air rate in kg/h

D = cyclone diameter in m

n = number of cyclones

ρ = density of air in kg/m3

ΔP= pressure drop across the cyclone in mm WG

K = cyclone constant

 

The cyclone constant depends upon the cyclone design and powder loading in air. For various types of the cyclones constants lay between 200 - 1000, the exact values being proprietary manufacturer know-how. However, if the air flow has been measured by one of the described methods and at the same time the pressure drop over the cyclone measured the cyclone constant can be calculated backwards and used in later routine measurements.

 

c) Measuring the amount of heat necessary for air heating:

 

If air is heated from a temperature t1 to a temperature t₂ the amount of air can be calculated from the amount of heat used and the temperature difference. In principle the same method of calculation is used for steam, oil, gas and electric air heaters.

 

The basic equation is:

where: 

E = efficiency of the heater

t₁ = air temperature at heater inlet

t₂ = air temperature at heater outlet

ca₁ = heat capacity of air at heater inlet

ca₂ = heat capacity of air at heater outlet

X = the heat consumed kcal/h.

 

Calculation of X for various types of heaters is explained below.

 

c1) Measuring of the condensate from the steam heater:

 

X for the equation [6,30] is:

where: 

W = amount of the condensate in kg/h

h_s = enthalpy of steam at heater inlet in kcal/kg

h_c = enthalpy of the condensate (which is equal to the temperature)

where: 

G = amount of oil in kg/h or gas in Nm³/h

Q_h = caloric value of the fuel in kcal/kg or kcal/Nm³.

 

d) Measuring the water evaporation rate:

 

The dryer is operated on water under constant t1 and t2 for at least 1 hour to obtain stable conditions. The amount of water, supplied to the dryer over a period of time is measured. The most suitable way is to measure the level difference h in a cylindrical feed tank of diameter D. The volume and weight of the evaporated water and the evaporation rate are:

where: 

D = feed tank diameter in m

h = level difference in m

t = time of measuring in s

ρw = density of water at feed temperature tf in °C

DER = rate of evaporation in kg/h.

 

The drying air rate A is then:

where: 

t₁ and t₂ = the air inlet and outlet temperatures,

ca₁ and ca₂ = heat capacities of air at t1 and t2,

cv₂ = heat capacity of water vapour at t2

t_f = feed temperature,

A_r = surface area of the dryer in m²,

t_s = spray dryer surrounding temperature,

K = radiation constant in kcal/m²/h.

where: 

p₀ = barometric pressure at sea level at 0°C,

M = molecular weight of air (= 0.029 kg/mol),

g = gravity constant (= 9.807 m/s2),

R = gas constant (=8.3144 J/K/mol)

T = absolute temperature in °K, and

m = altitude in m.

The spray dryer in operation is a system where air and product move through under changing temperatures and humidities and as the product is concerned, also changing physical properties. Entering components are: drying gas, which is usually heated ambient air, some auxiliary air flows (as cooling air, fines transport air etc.) and the feed to be dried. The humidity of the entering air corresponds to the ambient air humidity, possibly increased somewhat by moisture generated during combustion (in case of direct gas heating) or by moisture picked up by the air on passage through the building. The feed is the milk concentrate. Exhaust air and powder leave the system. The exhaust air is made up of all entering air flows plus water formed from the evaporation and besides it contains also some traces of the dried solids (fine particles). The dry products in powder form contain practically all the feed solids, but have residual moisture. The amount of air necessary to evaporate a required amount of water from a given amount of feed can be found by calculating all the individual heat requirements necessary for evaporating the water, heating or cooling each individual component from its inlet to its outlet temperature, while compensating for heat losses. The sum of these contributions is then recalculated into the air rate on the basis that the drying air enters the system with a temperature t₁ and leaves with a temperature t₂.

 

The following example is a demonstration of the calculation of the heat requirement and drying air rate of a spray dryer as specified below by points a) through g). The source of drying air is ambient thus the inlet humidity is considered as ambient humidity y_a:

 

 

Heat of evaporation:

Example: Calculate the drying air rate and size of dryer for following duty:

 

Feed rate: DFR = 4000 kg/h

Feed concentration: DFS = 48 %

Feed temperature: t_f = 60 °C

Product solids content: DPS = 95 %

Ambient temperature: t_a = 15 °C

Ambient humidity: y_a = 0.01 kg/kg

Inlet temperature: t₁ = 200 °C

Outlet temperature: t₂ = 80 °C

Cooling air rate: A_c = 200 kg/h

Fines transport air: A_t = 500 kg/h

Transport air temperature t_t = 60 °C

 

 

 

Total exhaust air = 39565.4 + 200 + 500 + 1978.9 = 42244.3 kg/h

Exhaust air density Vexair and volume Vexair are then:

The dryer should have two main cyclones of cyclone constant 380 operating at a pressure drop of 150 mm WG. The cyclone diameter according to the equation [6.29] will be: