[chenginst]

I had this problem while calculating the constant drying rate during flash/Pneumatic Drying. Drying rate is negative when I am calculating it.

During this type of drying, solid particle is freely flowing in a hot gas and free moisture evaporation occurs well below the boiling point near atmospheric pressure. Particle surface is saturated with water vapor.

First of all these are the conditions for drying and symbols used in equations.

T_s = Particle temperature = 303 K/ 30 °C
T_g = Gas temperature = 413 K/ 140 °C

P_sat = Saturation vapor pressure of water at particle temperature = 4247 Pa 
Source : https://www.engineeringtoolbox.com/water-vapor-saturation-pressure-d_599.html

P = atmospheric pressure = 101325 Pa

mole fraction of water vapor in gas is 0.15. Rest is dry gas.
P_H2O = Partial pressure of water vapor in gas = 101325 × 0.15 = 15198 Pa

M_w = molecular weight of water = 0.018 kg/mol

M_g = molecular weight of dry gas = 0.030 kg/mol

w = drying rate (kg/m²s)

k = mass transfer coefficient

H = humidity of gas ( H₂O kg / dry gas kg) =( 0.15 × 0.018 ) / ( 0.85 × 0.030 ) = 0.10588

R = universal gas constant = 8.314 J/mol K

In literature I found two equations to calculate the drying rate during flash/pneumatic drying.

This is the first one. It is for a particle.

I used trying \frac but it didn't work for me.

1.  w = k ( P_satM_w/RT_s - P_H2OM_w/RT_g )

However when I substitute the values the answer is negative. since 4247/303 - 15198/413 < 0

2. This is the second equation. This is for all the dispersed particles.

w =  k [ ( M_w P_sat / M_g(P - P_sat) ) - H ]

However when I substitute the values the answer is negative.

Since [18 × 4247 / 30 ×(101325 - 4247) ] - 0.10588  < 0

Can anyone please tell me why the drying rate is negative when it should be positive?

[mjc123]

The partial pressure of water in the gas is greater than the vapour pressure of water at the particle temperature, so water will condense rather than evaporate.

Are you sure you've got your numbers right? Particularly the mole fraction of water vapour in the gas - 0.15 seems rather high. That's very wet gas - not ideal for drying! (Was it perhaps 0.15%?)

[chenginst]

Actually drying gas is flue gas from burning hydrocarbons. Practically, the drying occurs.

The actual molar fraction of water vapor is 6%. But that doesn't make a difference. Drying rate is still negative.

[chenginst]

I found the reason why.

The gas temperature is 140 Celsius.

Although the drying rate is negative, water does not transfer from gas phase to solid phase since the dew point of gas at 6 percent mole fraction of water vapor (6 percent by volume) is around 40 Celsius.
Source: https://www.engineeringtoolbox.com/dew-point-flue-gases-d_1583.html

However, the particle temperature increases and Psat increases, the drying rate increases from negative to zero and then positive.

Hence, water won't evaporate from solid particle until drying rate is positive.